Precalculus : Algebraic Vectors and Parametric Equations

Example Questions

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Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Find the unit vector that is in the same direction as the vector

Explanation:

To find the unit vector in the same direction as a vector, we divide it by its magnitude.

The magnitude of  is .

We divide vector  by its magnitude to get the unit vector :

or

All unit vectors have a magnitude of , so to verify we are correct:

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

A unit vector has length .

Given the vector

find the unit vector in the same direction.

Explanation:

First, you must find the length of the vector. This is given by the equation:

Then, dividing the vector by its length gives the unit vector in the same direction.

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Put the vector  in unit vector form.

Explanation:

To get the unit vector that is in the same direction as the original vector , we divide the vector by the magnitude of the vector.

For , the magnitude is:

.

This means the unit vector in the same direction of  is,

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Find the unit vector of

.

Explanation:

In order to find the unit vector u of a given vector v, we follow the formula

Let

The magnitude of v follows the formula

.

For this vector in the problem

.

Following the unit vector formula and substituting for the vector and magnitude

.

As such,

.

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Find the unit vector of

Explanation:

In order to find the unit vector u of a given vector v, we follow the formula

Let

The magnitude of v follows the formula

For this vector in the problem

Following the unit vector formula and substituting for the vector and magnitude

As such,

Example Question #1 : Find A Vector Equation When Given Two Points

What is the vector that connects the point  to ?

Explanation:

The first step when solving for a vetor is to find the length of the vector. It can be helpful to visualize the system as a right triangle as seen below:

At this point, we can essentially solve the problem as if we're finding the hypotenuse and angle of a right triangle.

Using the Pythagorean Theorem to find the length of the vector we get:

Now we need to find the angle of the vector. We have all three sides of the triangle, so use the trig function that you are most comfortable with. The tangent function is used below because it uses sides that were given in the problem statement.

Example Question #2 : Find A Vector Equation When Given Two Points

Find the vector that starts at point  and ends at  and its magnitude.

Explanation:

To find the vector between two points, find the change between the points in the  and  directions, or  and . Then . If it helps, draw a line from the starting point to the end point on a graph and look at the changes in each direction.

We see that  and , so our vector is

To find a vectors magnitude, we sum up the squares of each component and take the square root:

Example Question #91 : Matrices And Vectors

Find the vector equation of the line through the points:

and .

Explanation:

The vector equation of the line through two points is the sum of one of the points and the direction vector between the two points scaled by a variable.

First we find the the direction vector by subtracting the two points:

.

Note that a line is continuous and defined on the real line. Then, we must scale the direction vector by a variable constant so as to define the line at each point. We then add one of the given points, so as to define the line through the given points. Either point can be chosen, but the correct answer uses the first point given.

Example Question #4 : Find A Vector Equation When Given Two Points

Given points  and , find a vector equation of the line passing these two points.

Explanation:

Write the formula to find the vector equation of the line.

Using  and , find the directional vector  by subtracting point A from B.

Substitute the directional vector  and point  into the formula.

A possible solution is:

Example Question #1 : Find A Direction Vector When Given Two Points

Find the directional vector of  if points A and B are  and , respectively.