Question

The vector has components +5 and +7 along the x- and y-axes respectively. along a set of axes rotated 90 degrees counterclockwise relative to the original axes, the vector's components are

Answer 1

Original Vector components are +5 in x-axis & +7 in y-axis,

After rotating 90 degrees counterclockwise,

Now, Vector components are +7 in x-axis & +5 in y-axis.

Answer 2

Refer to the diagram shown below.

Let the coordinates of the rotated vector be (a,b).
From the Pythagorean theorem,
d = √(5² + 7²) = 8.6023

The angle θ is given by
tan θ = 7/5 = 1.4
θ = tan⁻¹ 1.4 = 54.46°
φ = 180 - (90 + 54.46) = 35.54°

The coordinates of the rotated vector are
a = - d cos φ = - 8.6023*cos(35.54) = 7
b = d sin φ = 8.6023*sin(35.54) = 5

Answer:  (-7, 5)