Question

How does multiplying a vector by a scalar value of -pi / 4 change the vector?

Answer 1

Hello!

The answer is:

The second option:

The vector will change  direction and decrease in magnitude.

Why?

Multiplying a vector by a scalar will modify the magnitude of the vector, depending if the scalar is less or greater than "1" also, when a scalar has a negative sign, the direction of the new vector will be the opposite that the original vector.

Solving

We are given that the scalar value is equal to:

$-\frac{\pi }{4}=-0.79$

Let's use as original vector, the following vector:

$A=(2,2)$

Calculating its magnitude, we have:

$|A|=\sqrt{2^{2} +2^{2} }=2.83$

Then, multiplying the vector by the given scalar, we have:

$(-0.79A)=((-0.79)*2,-(0.79)*2)=(-1.58,-1.58)$

As we can see, the vector changed its direction, since both components are negative.

Calculating the magnitude of the new vector, we have:

$|A'|=\sqrt{(-1.58)^{2} +(-1.58)^{2} }=2.23$

We can see that the given scalar is less than "1", so the magnitude will decrease, also, the direction is the opposite of the original vector direction since both components have changed its sign.

Hence, we have that the magnitude of the new vector is less than the magnitude of the original vector, also, we can see that the direction of the new vector is the opposite of the original vector direction, so, the answer is the second option, the vector will change direction and decrease in magnitude.

Have a nice day!