You CAN Ace Calculus

 parametrics basics of vectors dot product cross product vector functions

### 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 > lines in 3-space

 Getting Started Parametric Equations Extended Parametric Discussion - Describing Lines Vector Function Symmetric Equations Practice

Up until now, you have worked with lines in two dimensions. It was convenient to define such lines using slope ($$m$$) and y-intercept ($$b$$) and write them in slope-intercept form $$y = mx + b$$. The only kind of lines that can not be written in this form are vertical lines, which we write as $$x = c$$. Another way to write the equation of a line is in general form $$Ax +By + C = 0$$. All lines in the plane can be written in this general form.

Now we are going to work with lines in three dimensions (sometimes called 3-space or just space).

When you studied lines in 2-dimensions, not only could you describe a line using the slope-intercept form, as mentioned above, but you could also describe it using parametric equations. An example might be to describe $$y=mx+b$$ in parametric form, we could use parameter t and write $$x = t$$ and $$y = mt + b$$.

At a minimum, a line in space is uniquely defined by two points on the line. However, there are several ways to specify lines in 3-dimensions. Let's use the graph on the right to get these forms. Here are a few things to notice about this graph.

- We want to specify the line that goes through the two points $$P_1$$ and $$P_2$$. The two points must be distinct to define a line, i.e. $$P_1 \neq P_2$$.
- The points $$P_1$$ and $$P_2$$ are known, i.e. we have the actual $$(x,y,z)$$ values for both of these points.

- Let $$P_1$$ be specified as $$P_1 = (x_1,y_1,z_1)$$.

- Let $$P_2$$ be specified as $$P_2 = (x_2,y_2,z_2)$$.

- The point P represents any point on the line and we write it as $$P = (x,y,z)$$. (note)

- Vector $$\vec{v}_1$$ (labeled v1 in the graph) is the vector from the origin to the point $$P_1$$. So $$\vec{v}_1 = \langle x_1, y_1, z_1 \rangle$$ or $$\vec{v}_1 = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$$.

- Vector $$\vec{v}_2$$ (labeled v2 in the graph) is the vector from the origin to the point $$P_2$$. So $$\vec{v}_2 = \langle x_2, y_2, z_2 \rangle$$ or $$\vec{v}_2 = x_2\hat{i} + y_2\hat{j} + z_2\hat{k}$$.

- Vector $$\vec{v}$$ is the vector from the origin to the point P. So $$\vec{v} = \langle x, y, z \rangle$$ or $$\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$$.

Here is a very good video clip explaining this idea and paralleling it with the slope-intercept form of a line that you already know.

### PatrickJMT - lines and planes [2mins-46secs]

video by PatrickJMT

Parametric Equations

One of the easiest ways to specify 3-dimensional lines is using parametric equations like we did for 2 dimensions. Using the above graph, let's derive one parametric equation. (There are an infinite number of sets of parametric equations.)

Define vector $$\vec{a}$$ as the vector from $$P_1$$ to $$P_2$$. We can write this as $$\vec{a} = \vec{v}_2 - \vec{v}_1$$. Defined this way, vector $$\vec{a}$$ ends up to be a vector parallel to the line.

Using the parameter t, any other point P on the line can be defined as $$\vec{v} = \vec{v}_1 + t \vec{a}$$. In this case, the domain of t is the set of all real numbers (t can be positive, negative or zero).

Specify vector $$\vec{a}$$ as $$\vec{a} = \langle a,b,c \rangle$$.
[Note: Using the letter a as both the vector name and as a scalar in the first component of the vector is not the best use of notation here. However, there should be no confusion since the vector is written with the vector arrow above the name, the scalar is written without it. This is an example of where notation is critically important. Mathematicians do this often, when the context and notation make it clear which a we are talking about.]

Using the equations $$\vec{v} = \vec{v}_1 + t \vec{a}$$ and $$\vec{a} = \langle a,b,c \rangle$$ we can write $$\langle x,y,z \rangle = \langle x_1,y_1,z_1 \rangle + t \langle a,b,c \rangle$$. If we equate the individual componenets we get the parametric equations

 $$x = x_1 + at$$ $$y = y_1 + bt$$ $$z = z_1 + ct$$

1. These equations are not unique since the vector $$\vec{a}$$ is dependent on the choices of $$\vec{v}_1$$ and $$\vec{v}_2$$.
2. We can emphasize that these parametric equations are functions of t by writing them as

 $$x(t) = x_1 + at$$ $$y(t) = y_1 + bt$$ $$z(t) = z_1 + ct$$

3. Another way to write the symmetric equations is as column vectors. This closely parallels the vector form $$\langle x,y,z \rangle = \langle x_1,y_1,z_1 \rangle + t \langle a,b,c \rangle$$.

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x_1 \\ y_2 \\ z_3 \end{bmatrix} + t \begin{bmatrix} a \\ b \\ c \end{bmatrix}$$

Extended Parametric Discussion - Describing Lines

Here is a video discussing various ways to describe a line, a line segment and a ray. The question he poses, and answers in the video, is this.
Let A and B be points with respective position vectors $$\vec{a}$$ and $$\vec{b}$$. Determine a parametric vector form for
1. the line segment AB;
2. the ray from B and passing through A;
3. all points P that lie on the line connecting A and B with A between P and B;
4. all points Q that lie on the line through A and B which lie closer to B than A.
This video will help you understand lines and vectors much more deeply.

### Dr Chris Tisdell - Equations of line segments and rays [13mins-56secs]

video by Dr Chris Tisdell

Vector Function

We can write a vector function using the parametric equations as $$\vec{V}(t) = (x_1+at)\hat{i} + (y_1+bt)\hat{j} + (z_1+ct)\hat{k}$$ or more simply as $$\vec{V}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$. Of course, since this vector function is dependent on non-unique parametric equations, there are many other (in fact, an infinite number) vector functions to describe any given line.

Symmetric Equations

Using the parametric equations above, we solve for t and equate them to get the symmetric equations as follows.

 $$\displaystyle{ x = x_1 + at ~~~ \to ~~~ t = \frac{x-x_1}{a} }$$ $$\displaystyle{ y = y_1 + bt ~~~ \to ~~~ t = \frac{y-y_1}{b} }$$ $$\displaystyle{ z = z_1 + ct ~~~ \to ~~~ t = \frac{z-z_1}{c} }$$

Since t is the same in all three equations, we know that the other side of the equations are equal. This gives us these symmetric equations.

$$\displaystyle{ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} }$$

1. Just as with the parametric equations, these equations are not unique but dependent on the choices of $$\vec{v}_1$$ and $$\vec{v}_2$$.
2. The values $$a,b,c$$ are called the direction numbers of the line. The name comes from the fact that the vector $$\vec{a} = \langle a,b,c \rangle$$ gives the direction of the line.
3. Of course, we are assuming here that $$a$$, $$b$$ and $$c$$ are all nonzero. If one of the values is zero, we can still write the symmetric equations and combine it with the corresponding parametric equation. If, for example, $$c=0$$, from the parametric equation $$z = z_1 + ct$$ we can see that $$z = z_1$$. So we write our symmetric equations as

 $$\displaystyle{ \frac{x-x_1}{a} = \frac{y-y_1}{b} }$$ $$z = z_1$$

Okay, time for some practice problems.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Find a vector equation for the line that goes through $$(1,3,2)$$ and $$(-4,3,0)$$.

Problem Statement

Find a vector equation for the line that goes through $$(1,3,2)$$ and $$(-4,3,0)$$.

Solution

### 1251 solution video

video by PatrickJMT

Find the symmetric equations of a line through the point $$P(2,3,-4)$$ in the same direction as the vector $$\vec{v} = \langle 1,-1,-2 \rangle$$.

Problem Statement

Find the symmetric equations of a line through the point $$P(2,3,-4)$$ in the same direction as the vector $$\vec{v} = \langle 1,-1,-2 \rangle$$.

Solution

### 1257 solution video

video by Krista King Math

Find the parametric equations of the line that passes through the point $$P(-1,2,3)$$ and is parallel to the vector $$\vec{v}=\langle4,5,6\rangle$$.

Problem Statement

Find the parametric equations of the line that passes through the point $$P(-1,2,3)$$ and is parallel to the vector $$\vec{v}=\langle4,5,6\rangle$$.

Solution

### 1852 solution video

video by PatrickJMT

Find the scalar equation of the line through the point $$P(0,0,0)$$ in the direction $$\vec{v} = \hat{i}+2\hat{j}+3\hat{k}$$.

Problem Statement

Find the scalar equation of the line through the point $$P(0,0,0)$$ in the direction $$\vec{v} = \hat{i}+2\hat{j}+3\hat{k}$$.

Solution

### 1258 solution video

video by Krista King Math

Find the scalar equation of the line through the point $$P(4,13,-3)$$ in the direction $$\vec{v} = 2\hat{i}-3\hat{k}$$.

Problem Statement

Find the scalar equation of the line through the point $$P(4,13,-3)$$ in the direction $$\vec{v} = 2\hat{i}-3\hat{k}$$.

Solution

### 1259 solution video

video by Krista King Math

Find the scalar equation of the line through the points $$P(0,0,0)$$ and $$Q(-6,3,5)$$.

Problem Statement

Find the scalar equation of the line through the points $$P(0,0,0)$$ and $$Q(-6,3,5)$$.

Solution

### 1260 solution video

video by Krista King Math

Find the scalar equation of the line through the points $$P(3,5,7)$$ and $$P(6,5,4)$$.

Problem Statement

Find the scalar equation of the line through the points $$P(3,5,7)$$ and $$P(6,5,4)$$.

Solution

### 1261 solution video

video by Krista King Math