# Precalculus : Find the Product of a Vector and a Scalar

## Example Questions

### Example Question #1 : Find The Product Of A Vector And A Scalar

Find the product of the vector  and scalar .

Explanation:

When multiplying a vector by a scalar we multiply each component of the vector by the scalar and the result is a vector:

### Example Question #2 : Find The Product Of A Vector And A Scalar

Find the vector given by the product:

Explanation:

Given a scalar k and a vector v, the vector given by their products is defined component-wise:

.

Here, our product is:

### Example Question #3 : Find The Product Of A Vector And A Scalar

This question refers to the previous question.

Simplify.

Explanation:

In order to simplify this problem we need to multiply the scalar factor to each component of the vector.

In our case the scalar factor is

Thus,

### Example Question #4 : Find The Product Of A Vector And A Scalar

Evaluate:

Explanation:

In order to determine the final value of the vector, distribute the scalar among each term in the vector.

### Example Question #5 : Find The Product Of A Vector And A Scalar

If the vector from  to  was multiplied by a scale factor of 3, what is the new vector ?

Explanation:

To find :

Subtract vector  from .

Multiply this vector by a scale of 3.

### Example Question #6 : Find The Product Of A Vector And A Scalar

Find the product of:

Explanation:

When a scalar is multiplied to a vector, simply distribute that value for both terms in the vector.

### Example Question #7 : Find The Product Of A Vector And A Scalar

When given a vector  and a scalar  what happens to the length and angle of  when multiplied with ?

or the length of the product is the same as the original vector.

The angle is unchanged

or the length of the product is  times the length of the original vector.

The angle is unchanged.

or the length of the product is the same as the original vector.

The angle is multiplied by .

or the length of the product is  times as long as the original vector.

The angle is multiplied by .

or the length of the product is  times the length of the original vector.

The angle is multiplied by

or the length of the product is  times the length of the original vector.

The angle is unchanged.

Explanation:

In simple terms  is the hypotenuse of a triangle formed by the components of . So when you multiply  by  it mupltiplies all the componets by . This makes the length of the hypotensuse grow by  as demonstrated by  from the Pythagorean Theorem.

For the same reasons the angle does not change because the new longer triangle will be a similar triangle to the original triangle.

### Example Question #8 : Find The Product Of A Vector And A Scalar

Determine the product: