# Calculus 2 : Vector Calculations

## Example Questions

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### Example Question #1 : Vector Calculations

Find the cross product of the following two vectors:

Possible Answers:

Correct answer:

Explanation:

We obtain the cross product of the two vectors by setting up the following matrix:

Where our first row represents the unit vectors, the second row represents vector a, and the third row represents vector b. The first component of our cross product is obtained by taking the determinant of the matrix left by crossing out the row and column in which  is located. Accordingly, our second and third components are found by taking the determinant of the matrix left by crossing out the rows and columns in which    and    are located, respectively. This process gives us the following simple equation for expressing the cross product of two vectors, into which we can plug in the components of our vectors to find the cross product:

### Example Question #2 : Vector Calculations

Evaluate the dot product:

Possible Answers:

Correct answer:

Explanation:

Let vector   and .

The dot product is equal to:

Following this rule for the current problem, simplify.

### Example Question #3 : Vector Calculations

Evaluate:

Possible Answers:

Correct answer:

Explanation:

Let vector   and .

Use the following formula to solve this dot product:

Substitute and solve.

### Example Question #4 : Vector Calculations

Calculate the cross product:

Possible Answers:

Correct answer:

Explanation:

The vectors can be rewritten in the following form:

One method of solving the 3 by 3 determinant is to break this down into 2 by 2 determinants. The determinant of 2 by 2 matrices are computed by , such that:

Rewrite the 3 by 3 determinant.

### Example Question #5 : Vector Calculations

A sling shoots a rock  feet per second at an elevation angle of  degrees. What are the horizontal and vertical components in vector form?

Possible Answers:

Correct answer:

Explanation:

The horizontal and vertical components are shown below:

Plug in the velocity and the given angle to the equations.

Therefore, the components in vector form is .

### Example Question #6 : Vector Calculations

Evaluate the dot product of  and .

Possible Answers:

Correct answer:

Explanation:

Let vectors   and .

The formula for the dot product is:

Follow this formula and simplify.

### Example Question #7 : Vector Calculations

Solve:

Possible Answers:

Correct answer:

Explanation:

The problem  is in the form of a dot product.  The final answer must be an integer, and not in vector form.

Write the formula for the dot product.

Substitute the givens and solve.

### Example Question #8 : Vector Calculations

Suppose .  Find the magnitude of .

Possible Answers:

Correct answer:

Explanation:

Calculate .

Find the magnitude.

### Example Question #9 : Vector Calculations

Two particles move freely in two dimensional space. The first particle's location as a function of time is , and the second particle's location is . Will the particles ever collide for

Possible Answers:

Impossible to determine

Yes, because the particles have the same  or  component (not necessarily simultaneously).

Yes, because the particles'  and  coordinates are the same simultaneously at a certain instant in time.

No, because the particles never have the same  or  component (not necessarily simultaneously).

No, because the particles'  and  coordinates are never the same simultaneously at any instant in time.

Correct answer:

No, because the particles'  and  coordinates are never the same simultaneously at any instant in time.

Explanation:

In order for the particles to collide, their  and  coordinates must be equal simultaneously. In order to check if this happens, we can set the particles' -coordinates and -coordinates equal to each other.

Let's start with the -coordinate:

.

Using the quadratic formula

, we get

the other root is negative, so it can be discarded since .

Now let's do the -coordinate: . Use the quadratic formula again to solve for , and you'll get  (these roots are approximately  and ). The particles never have the same  and  coordinate simultaneously, so they do not collide.

### Example Question #10 : Vector Calculations

Calculate

Possible Answers:

Correct answer:

Explanation:

is simply the dot product of these two vectors. Mathematically, this is calculated as follows.

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