Part A of this question asks for the sum of vector A = 3x+5y and vector B = 1x-3y. I did this and got the vector A + vector B = 4x+2y.

Part B then asks what the magnitude and direction of vector A + vector B? The answer in the back of the textbook says that it is 4.5 at a 27 degree angle, but how do you get to this??

from the origin, 4 to the right and 2 up

so
tan (angle) = 2/4 = .5
so
angle = 26.57 degrees

sqrt (4^2 + 2^2) = sqrt (20) = 4.47

To find the magnitude and direction of vector A + vector B, you need to calculate the magnitude and direction separately.

Magnitude:
The magnitude of a vector can be found using the Pythagorean theorem. Given vector A + vector B = 4x + 2y, the magnitude can be calculated as follows:

Magnitude = √((4)^2 + (2)^2)
= √(16 + 4)
= √20
≈ 4.47

Therefore, the magnitude of vector A + vector B is approximately 4.47.

Direction:
The direction of a vector can be represented using an angle. To find the direction of vector A + vector B, you can use trigonometric functions.

In this case, you need to find the angle (θ) that the resultant vector (vector A + vector B) makes with the positive x-axis. You can use the inverse tangent function to determine this angle.

Direction (θ) = tan^(-1)((2)/(4))
≈ 26.57 degrees

So, the direction of vector A + vector B is approximately 26.57 degrees.

However, since you mentioned that the answer in the textbook states it as 4.5 at a 27-degree angle, it's possible that there might be a calculation error or rounding difference. Double-check your calculations to make sure you've used the correct values.