Answer to Question #331638 in Linear Algebra for Athar

Question

 Suppose that a vector a in the xy-plane points in a direction that is 47◦ counterclockwise from the positive x-axis, and a vector b in that plane points in a direction that is 43◦ clockwise from the positive x-axis. What can you say about the value of a · b?

Answer 1

If a vector a in the xy-plane points in a direction that is 47° counterclockwise from the positive x-axis, and a vector b in that plane points in a direction that is 43° clockwise from the positive x-axis, than the angle between a and b would be equal to 47° + 43° = 90°.

a • b means the Dot Product of a and b.

We can calculate the Dot Product of two vectors this way:

a · b = |a| × |b| × cos(θ)

where:

|a| is the magnitude (length) of vector a

|b| is the magnitude (length) of vector b

θ is the angle between a and b

We already know that the angle between a and b is 90°.

When two vectors are at right angles to each other the dot product is zero because cos 90° = 0.

So the answer is: the dot product of the vectors a and b equals to zero (a • b = 0).