You CAN Ace Calculus

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

17calculus > vectors > dot product

The dot product is one of two main ways we 'multiply' vectors (the other way is the cross product). We call this 'multiplication' a dot product since we write the dot product using a 'dot' between the vectors. This dot is usually a solid dot like $$\cdot$$ instead of an open dot (or circle) like $$\circ$$, which is usually used for the composite of two functions.

### Dot Product Study Hints

2. Notice the notation used in your textbook and on this site.
3. Make flashcards (handwritten is best but a system like quizlet will also work) to help prepare for exams. Study your cards on the bus, between classes or while waiting for class to start.
4. Check for additional general study hints on the study techniques page.

If we have vectors, $$\vec{u} = \langle u_1, u_2, u_3 \rangle$$ and $$\vec{v} = \langle v_1, v_2, v_3 \rangle$$, the dot product is $$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$$.
With dot products, we are required to write the dot between $$\vec{u}$$ and $$\vec{v}$$ to indicate the dot product. Writing the two vectors side by side with nothing in between like this $$\vec{u} \vec{v}$$ makes no sense and is incorrect. In two dimensions we can think of $$u_3 = 0$$ and $$v_3 = 0$$ and the above equation holds. Notice the result of the dot product of two vectors is a scalar. This is why the dot product is sometimes called the scalar product.

Okay, so let's watch a video clip showing a quick overview of the dot product. This video is for two dimensional vectors.

### PatrickJMT - overview of the dot product [1min-37secs]

video by PatrickJMT

It works the same for three dimensional vectors, shown in this video clip.

### PatrickJMT - dot product in three dimensions [23secs]

video by PatrickJMT

Properties of the Dot Product

If we let $$\vec{u}$$, $$\vec{v}$$ and $$\vec{w}$$ be vectors and $$k$$ be a scalar, the following properties hold for the dot product.

$$\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$$

commutative property

$$\vec{u} \cdot ( \vec{v} + \vec{w} ) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w}$$

distributive property

$$k(\vec{u} \cdot \vec{v} ) = k\vec{u} \cdot \vec{v} = \vec{u} \cdot k\vec{v}$$

$$\vec{u} \cdot \vec{0} = 0$$

$$\vec{0}$$ is the zero vector

$$\vec{v} \cdot \vec{v} = \norm{\vec{v}} ^2$$

The double bars on $$\norm{\vec{v}}$$ indicate the magnitude of the vector $$\vec{v}$$, usually called the vector norm. See the Vector Representations panel on the main vectors page for detail on the vector norm.

### Larson Calculus - properties proofs [2mins-34secs]

video by Larson Calculus

Before we go on to look at some applications of the dot product, let's work a few practice problems.

Instructions - - Unless otherwise instructed, calculate the dot product of each set of vectors giving your answers in exact form. For angles, give your answers in radians to 3 decimal places.

$$\vec{a}=\langle 2,5\rangle$$, $$\vec{b}=\langle-3,1\rangle$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{a}=\langle 2,5\rangle$$, $$\vec{b}=\langle-3,1\rangle$$

Solution

### 1237 solution video

video by PatrickJMT

$$\vec{u} = 2\vec{i} + \vec{j}-\vec{k}$$, $$\vec{v} = \vec{i} + 7\vec{j}$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{u} = 2\vec{i} + \vec{j}-\vec{k}$$, $$\vec{v} = \vec{i} + 7\vec{j}$$

$$\vec{u} \cdot \vec{v} = 9$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{u} = 2\vec{i} + \vec{j}-\vec{k}$$, $$\vec{v} = \vec{i} + 7\vec{j}$$

Solution

Even though the second vector looks like a two dimensional vector, we can assume that the missing coordinate is zero giving us 2 three dimensional vectors. Otherwise the question would not make sense.

 $$\vec{u} \cdot \vec{v}$$ $$\left( 2\vec{i} + \vec{j} - \vec{k}\right) \cdot \left( \vec{i} + 7\vec{j} + 0\vec{k}\right)$$ $$2(1) + 1(7) - 1(0)$$ $$2 + 7 - 0 = 9$$

$$\vec{u} \cdot \vec{v} = 9$$

$$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{B} = \hat{i} + 3\hat{k}$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{B} = \hat{i} + 3\hat{k}$$

$$(2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (\hat{i} + 3\hat{k}) = 14$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{B} = \hat{i} + 3\hat{k}$$

Solution

 $$\vec{A} \cdot \vec{B}$$ $$(2\hat{i}+3\hat{j}+4\hat{k}) \cdot (\hat{i}+3\hat{k})$$ $$2(1) + 3(0) + 4(3) = 2+0+12 = 14$$

$$(2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (\hat{i} + 3\hat{k}) = 14$$

$$\vec{u}=2\vec{i}+4\vec{j}-17\vec{k}$$, $$\vec{v}=\vec{i}+5\vec{j}+\vec{k}$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{u}=2\vec{i}+4\vec{j}-17\vec{k}$$, $$\vec{v}=\vec{i}+5\vec{j}+\vec{k}$$

$$\vec{u}\cdot\vec{v}=5$$

Problem Statement

Calculate the dot product of the two vectors $$\vec{u}=2\vec{i}+4\vec{j}-17\vec{k}$$, $$\vec{v}=\vec{i}+5\vec{j}+\vec{k}$$

Solution

 $$\vec{u} \cdot \vec{v}$$ $$\left( 2\vec{i} + 4\vec{j} - 17\vec{k}\right) \cdot \left( \vec{i} + 5\vec{j} + \vec{k} \right)$$ $$2(1) + 4(5) - 17(1)$$ $$2 + 20 - 17 = 5$$

$$\vec{u}\cdot\vec{v}=5$$

Applications of the Dot Product

Here are a few applications where we can use the dot product to solve specific problems. More applications can be found on the cross products page.

Angle Between Vectors

Using the Law of Cosines, we show that $$\displaystyle{ \cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{ \norm{\vec{u}} \norm{\vec{v}} } }$$ where $$\theta$$ is the angle between the vectors $$\vec{u}$$ and $$\vec{v}$$.

We can use this equation as an alternate way to calculate the dot product as follows.
$$\vec{u} \cdot \vec{v} = \cos (\theta) \norm{\vec{u}} \norm{\vec{v}}$$

### Angle Between Two Vectors Proof

Theorem: Angle Between Two Vectors

If $$\theta$$ is the angle between two nonzero vectors $$\vec{u}$$ and $$\vec{v}$$, then $\cos\theta = \frac{\vec{u} \cdot \vec{v}}{\norm{\vec{u}} \norm{\vec{v}} }$

Proof Overview - - We will use the triangle on the right and the Law of Cosines to prove this theorem.
The Law of Cosines tells us $$\norm{\vec{a}}^2 = \norm{\vec{b}}^2 + \norm{\vec{c}}^2 - 2\norm{\vec{b}} \norm{\vec{c}} \cos(\theta)$$. In terms of these vectors, our goal equation is $\cos(\theta) = \frac{\vec{b} \cdot \vec{c}}{\norm{\vec{b}} \norm{\vec{c}}}$ Since our goal equation does not have $$\norm{\vec{a}}$$ in it, let's look at the triangle to see if we can find a relationship between the vector $$\vec{a}$$ and the other two vectors.

From basic vector addition, we see from the triangle that $$\vec{c} + \vec{a} = \vec{b}$$. Solving for vector $$\vec{a}$$, we have $$\vec{a} = \vec{b} - \vec{c}$$. We can take the norm and square both sides(1) to get $$\norm{\vec{a}}^2 = \norm{\vec{b} - \vec{c}}^2$$. Let's look closer at the right side of this equation.

$$\begin{array}{rcl} \norm{\vec{b} - \vec{c}}^2 & = & \left( \vec{b} - \vec{c} \right) \cdot \left( \vec{b} - \vec{c} \right) \\ & = & \vec{b} \cdot \left( \vec{b} - \vec{c} \right) - \vec{c} \cdot \left( \vec{b} - \vec{c} \right) \\ & = & \vec{b} \cdot \vec{b} - \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} \\ & = & \norm{\vec{b}}^2 - 2\vec{b} \cdot \vec{c} + \norm{\vec{c}}^2 \end{array}$$

Since this expression is equal to $$\norm{\vec{a}}^2$$, we can substitute this result for $$\norm{\vec{a}}^2$$ in the Law of Cosines equation above.

$$\begin{array}{rcl} \norm{\vec{b}}^2 - 2\vec{b} \cdot \vec{c} + \norm{\vec{c}}^2 & = & \norm{\vec{b}}^2 + \norm{\vec{c}}^2 - 2\norm{\vec{b}} \norm{\vec{c}} \cos(\theta) \\ - 2\vec{b} \cdot \vec{c} & = & - 2\norm{\vec{b}} \norm{\vec{c}} \cos(\theta) \\ \cos(\theta) & = & \frac{\vec{b} \cdot \vec{c} }{\norm{\vec{b}} \norm{\vec{c}}} \text{ [qed] } \end{array}$$

Notes - - -
(1) Be very careful when you square both sides of an equation. It is strictly true only when both sides are guaranteed to be positive. If one side could be negative, we have lost that information if we need to go back to it later. Since the norm is always positive, we can safely do it here with no fear of losing information.

Now, let's stop and think for a moment.

orthogonal

Perpendicular, orthogonal and normal all essentially mean the same thing - meeting at right angles (90 degrees, π/2 radians). But in mathematics we say that two vectors are orthogonal, two lines or planes are perpendicular and a vector is normal to a line or plane.

 What is the dot product when the vectors are orthogonal? Think about it for a minute and then click here for the answer

### Larson Calculus - proof of orthogonal vector [2mins-28secs]

video by Larson Calculus

Here are some practice problems involving angles between vectors and orthogonal vectors.

Calculate the angle between the vectors $$\vec{v} = 2\vec{i} + 3\vec{j} + 1\vec{k}$$ and $$\vec{w} = 4\vec{i} + 1\vec{j} + 2\vec{k}$$.

Problem Statement

Calculate the angle between the vectors $$\vec{v} = 2\vec{i} + 3\vec{j} + 1\vec{k}$$ and $$\vec{w} = 4\vec{i} + 1\vec{j} + 2\vec{k}$$.

The angle between the vectors $$\vec{v}=2\vec{i}+3\vec{j}+1\vec{k}$$ and $$\vec{w}=4\vec{i}+1\vec{j}+2\vec{k}$$ is $$\displaystyle{ \arccos\left(\frac{13}{7\sqrt{6}} \right) \approx 0.710}$$ radians.

Problem Statement

Calculate the angle between the vectors $$\vec{v} = 2\vec{i} + 3\vec{j} + 1\vec{k}$$ and $$\vec{w} = 4\vec{i} + 1\vec{j} + 2\vec{k}$$.

Solution

To calculate the angle between the two vectors, we use the equation $$\displaystyle{ \cos(\theta ) = \frac{\vec{v} \cdot \vec{w}}{\| \vec{v} \| \| \vec{w} \|} }$$.
First, let's find the dot product.
$$\vec{v} \cdot \vec{w} = \left( 2\vec{i} + 3\vec{j} + 1\vec{k} \right) \cdot \left( 4\vec{i} + 1\vec{j} + 2\vec{k} \right) = 2(4) + 3(1) + 1(2) = 13$$
Now we need to find the norm (magnitude) of each of the vectors.
$$\| \vec{v} \| = \sqrt{ 2^2 + 3^2 + 1^2} = \sqrt{4+9+1} = \sqrt{14}$$
$$\| \vec{w} \| = \sqrt({ 4^2 + 1^2 + 2^2} = \sqrt{16+1+4} = \sqrt{21}$$
So now we have $$\displaystyle{ \cos(\theta) = \frac{13}{\sqrt{14}\sqrt{21}} = \frac{13}{\sqrt{2(7)}\sqrt{3(7)}} = \frac{13}{7\sqrt{6}} }$$
Solving for $$\theta$$ we get $$\displaystyle{ \theta = \arccos\left(\frac{13}{7\sqrt{6}} \right) \approx 0.710286 }$$

The angle between the vectors $$\vec{v}=2\vec{i}+3\vec{j}+1\vec{k}$$ and $$\vec{w}=4\vec{i}+1\vec{j}+2\vec{k}$$ is $$\displaystyle{ \arccos\left(\frac{13}{7\sqrt{6}} \right) \approx 0.710}$$ radians.

Find the angle between the vectors $$\vec{a} = 6\hat{i} - 2\hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$.

Problem Statement

Find the angle between the vectors $$\vec{a} = 6\hat{i} - 2\hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$.

Solution

### 1238 solution video

video by PatrickJMT

Calculate the angle between $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$\vec{B} = \hat{i} + 3\hat{k}$$.

Problem Statement

Calculate the angle between $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$\vec{B} = \hat{i} + 3\hat{k}$$.

The angle between $$\vec{A}=2\hat{i}+3\hat{j}+4\hat{k}$$ and $$\vec{B}=\hat{i}+3\hat{k}$$ is $$\displaystyle{\arccos\left( \frac{14}{\sqrt{290}} \right) \approx 0.606 }$$ radians.

Problem Statement

Calculate the angle between $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$\vec{B} = \hat{i} + 3\hat{k}$$.

Solution

From a previous practice problem, we know that that dot product of these vectors is $$14$$. In order to find the angle between the vectors, we can use the formula $$\vec{A} \cdot \vec{B} = \|\vec{A}\| \|\vec{B}\|\cos(\theta)$$ where $$\theta$$ is the angle between the two vectors.
Let's find the magnitudes of the two vectors.
$$\|\vec{A}\| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4+9+16} = \sqrt{29}$$
$$\|\vec{B}\| = \sqrt{1^2+0^2+3^2} = \sqrt{1+9} = \sqrt{10}$$
So now we have

 $$\displaystyle{ \cos(\theta) = \frac{\vec{A}\cdot\vec{B}}{\|\vec{A}\| \|\vec{B}\|} }$$ $$\displaystyle{ \cos(\theta) = \frac{14}{(\sqrt{29})(\sqrt{10})} }$$ $$\displaystyle{ \theta = \arccos\left( \frac{14}{\sqrt{290}} \right) \approx 0.606 }$$

The angle between $$\vec{A}=2\hat{i}+3\hat{j}+4\hat{k}$$ and $$\vec{B}=\hat{i}+3\hat{k}$$ is $$\displaystyle{\arccos\left( \frac{14}{\sqrt{290}} \right) \approx 0.606 }$$ radians.

Find the angle between the vectors $$\vec{a} = \langle 1,2,3 \rangle$$ and $$\vec{b} = \langle -3,-1,4 \rangle$$.

Problem Statement

Find the angle between the vectors $$\vec{a} = \langle 1,2,3 \rangle$$ and $$\vec{b} = \langle -3,-1,4 \rangle$$.

Solution

### 1245 solution video

video by PatrickJMT

Are the vectors $$\vec{a}=\langle2,4\rangle$$ and $$\vec{b}=\langle4,-2\rangle$$ orthogonal?

Problem Statement

Are the vectors $$\vec{a}=\langle2,4\rangle$$ and $$\vec{b}=\langle4,-2\rangle$$ orthogonal?

Solution

### 1239 solution video

video by PatrickJMT

Find the angle between the vectors $$\langle 5,2 \rangle$$ and $$\langle 3,4 \rangle$$.

Problem Statement

Find the angle between the vectors $$\langle 5,2 \rangle$$ and $$\langle 3,4 \rangle$$.

Solution

### 1807 solution video

video by Larson Calculus

Find the angle between the vectors $$\vec{a} = \langle 2,3,5 \rangle$$ and $$\vec{b} = \langle 1,6,-4 \rangle$$.

Problem Statement

Find the angle between the vectors $$\vec{a} = \langle 2,3,5 \rangle$$ and $$\vec{b} = \langle 1,6,-4 \rangle$$.

Solution

### 1246 solution video

video by PatrickJMT

Find the angle between the vectors $$\langle4,6\rangle$$ and $$\langle3,-2\rangle$$.

Problem Statement

Find the angle between the vectors $$\langle4,6\rangle$$ and $$\langle3,-2\rangle$$.

$$\pi/2$$ radians

Problem Statement

Find the angle between the vectors $$\langle4,6\rangle$$ and $$\langle3,-2\rangle$$.

Solution

In the video, he gives his answer in degrees. However, in calculus we usually use radians.

### 1808 solution video

video by Larson Calculus

$$\pi/2$$ radians

Direction Cosines and Direction Angles

Direction angles are the angles between a given vector $$\vec{v}$$ and each coordinate axis (usually in three dimensions, so there are three of them). Basically, we use the equation for the angle between vectors to get the direction cosine equations and the direction angles. For example, to find the direction cosine and the direction angle between a vector $$\vec{v}$$ and the x-axis, we have
$$\displaystyle{ \cos(\alpha) = \frac{\vec{v} \cdot \hat{i}}{\norm{\vec{v}} \norm{\hat{i}}} }$$
Let's label the components of $$\vec{v}$$ as $$\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$$
Since $$\hat{i}$$ is the unit vector in the direction of the x-axis, we can write $$\hat{i} = 1\hat{i} + 0\hat{j} + 0\hat{k}$$ and $$\|\hat{i}\| = 1$$.
$$\displaystyle{ \cos(\alpha) = \frac{\vec{v} \cdot \hat{i}}{\|\vec{v}\| \|\hat{i}\|} = \frac{v_1}{\|\vec{v}\|} }$$
Similar results can be obtained for the other two angles. Most textbooks and mathematicians use special greek letters for these angles as listed below.

Angle Description Direction Cosine Direction Angle $$\alpha$$ is the angle between $$\vec{v}$$ and $$\hat{i}$$ $$\displaystyle{ \cos(\alpha) = \frac{v_1}{\|\vec{v}\|} }$$ $$\displaystyle{ \alpha = \arccos\left(\frac{v_1}{\|\vec{v}\|} \right) }$$ $$\beta$$ is the angle between $$\vec{v}$$ and $$\hat{j}$$ $$\displaystyle{ \cos(\beta) = \frac{v_2}{\|\vec{v}\|} }$$ $$\displaystyle{ \beta = \arccos\left(\frac{v_2}{\|\vec{v}\|} \right) }$$ $$\gamma$$ is the angle between $$\vec{v}$$ and $$\hat{k}$$ $$\displaystyle{ \cos(\gamma) = \frac{v_3}{\|\vec{v}\|} }$$ $$\displaystyle{ \gamma = \arccos\left(\frac{v_3}{\|\vec{v}\|} \right) }$$

Study Hint: Since you already need to know the equation for the angle between two vectors, just remember what the direction cosines and direction angles are. You can derive the equations in the above table from that information. Additionally, if you just memorize the equations, you may not remember what they represent or where they come from. What are you going to do when your instructor asks you to define they mean and where they come from?

Find the direction angle of $$3 \vhat{i} - 4\vhat{j}$$.

Problem Statement

Find the direction angle of $$3 \vhat{i} - 4\vhat{j}$$.

Solution

### 1809 solution video

video by Larson Calculus

Projection and Vector Components

Projection
We know from basic trigonometry, the projection of vector $$\vec{A}$$ onto vector $$\vec{B}$$ (see figure to the right) is just $$\|\vec{A}\| \cos(\theta)$$. However, what if we don't know the value of angle $$\theta$$? We can use the dot product to find the projection of one vector onto another. Let's derive the equation.

From the picture on the right, it is easy to see that the magnitude of the projection of vector $$\vec{A}$$ onto vector $$\vec{B}$$ is $$\|proj_{\vec{B}} \vec{A} \| = \|\vec{A}\| \cos(\theta)$$. Since we don't know the angle $$\theta$$, we can use the angle between vectors formula to substitute for $$\cos(\theta)$$ as follows.

$$\begin{array}{rcl} \| proj_{\vec{B}} \vec{A}\| & = & \|\vec{A}\| \cos(\theta) \\ & = & \displaystyle{ \|\vec{A}\| \frac{\vec{A} \cdot \vec{B}}{\|\vec{A}\| \|\vec{B}\|} } \\ & = & \displaystyle{ \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|} } \end{array}$$

Notice that this result is just the magnitude of the projection of vector $$\vec{A}$$ onto $$\vec{B}$$. If we want to find the projection vector itself, we can multiply this result by the unit vector in the direction of $$\vec{B}$$ and end up with

$$\displaystyle{ proj_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|} \frac{\vec{B}}{\|\vec{B}\|} = \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|^2} \vec{B} }$$

This last equation is the preferred equation since calculating $$\|\vec{B}\|$$ requires taking a square root, so squaring it removes that difficulty.

Here is a great video explaining vector projection in more detail.

### Dr Chris Tisdell - vector projection [17mins-55secs]

video by Dr Chris Tisdell

Vector Component
Now that we have the projection vector, it is a simple matter to find the vector component of $$\vec{A}$$ orthogonal to $$\vec{B}$$ (the dashed line in the above picture) using basic addition of vectors. Let's call our unknown vector $$\vec{v}$$. We know that $$proj_{\vec{B}} \vec{A} + \vec{v} = \vec{A}$$. Solving for $$\vec{v}$$, we have the vector component of $$\vec{A}$$ that is orthogonal to $$\vec{B}$$ which is

$$\vec{v} = \vec{A} - proj_{\vec{B}} \vec{A}$$.

Study Note - - We highly recommend that you spend some going through this section carefully until you understand it well. When you get to lines and planes, you will need to understand and be able to use this concept.

Notation Note: You may find different notation for the projection vector than we used here. The notation $$proj_{\vec{B}} \vec{A}$$ is fairly common but not used exclusively.

Find the projection of $$\vec{u}=\langle 25, 25\sqrt{3} \rangle$$ onto $$\vec{v}=\langle 11,4 \rangle$$.

Problem Statement

Find the projection of $$\vec{u}=\langle 25, 25\sqrt{3} \rangle$$ onto $$\vec{v}=\langle 11,4 \rangle$$.

$$\langle 36,13 \rangle$$

Problem Statement

Find the projection of $$\vec{u}=\langle 25, 25\sqrt{3} \rangle$$ onto $$\vec{v}=\langle 11,4 \rangle$$.

Solution

These videos are part of a longer video and this is just part of the example. You do not need to know the missing parts in order to extract the solution. The explanation is shown in three video clips.

### 1810 solution video

video by Larson Calculus

### 1810 solution video

video by Larson Calculus

### 1810 solution video

video by Larson Calculus

$$\langle 36,13 \rangle$$

Calculating Work

One of the major applications of the dot product is to calculate work.

Work is defined as the magnitude of a force acting on an object times the distance the object moves. Force is a vector and the only part of the vector that contributes to the work is the part in the direction the object moves. So, if we define a vector $$\vec{d}$$ that points in the direction that the object moves whose magnitude is the distance moved from point $$P$$ to point $$Q$$ (see the figure to the right), the work, $$W$$, is

$$W = \| proj_{\vec{d}} \vec{F} \| \| \vec{d} \|$$

Now, the length of the projection of vector $$\vec{F}$$ onto vector $$\vec{d}$$ is $$\| proj_{\vec{d}} \vec{F} \| = \|\vec{F}\| \cos(\theta)$$
So the work equation becomes $$W = \|\vec{F}\| \cos(\theta) \| \vec{d} \| = \vec{F} \cdot \vec{d}$$.

Here we have shown there are two (equivalent) equations to calculate the work.

$$W = \| proj_{\vec{d}} \vec{F} \| \| \vec{d} \|$$

$$W = \vec{F} \cdot \vec{d}$$

Your next logical step is the vector cross product.