Question

A certain vector in the XY plane has an x component of 4m and a y component of 10 m. It is then rotated in the XY plane so that its x-component is doubled. Then its new y component is (approximately):
A. 20 m
B. 7.2 m
C. 5.0 m
D. 4.5 m

Answer 1

Hint: In this question, we need to determine the new y- component of the vector such that the x-component gets its absolute value doubled. For this, we need to follow that concept that the magnitude of the resultant vector will remain the same whenever the vectors are rotated irrespective of the direction.
If the vector is given whose x and y coordinate are also given, and this vector is rotated along the XY plane, we know that upon rotation of the vector their magnitude remains same hence first find the magnitude of the vector before rotation and then find y coordinate with the help of this magnitude.

Complete step by step answer:
X-component \[ = 4m\]
Y-component \[ = 10m\]
Let the vector be A whose magnitude is given as
\[
  {A^2} = x_1^2 + y_1^2 \\
   = {4^2} + {10^2} \\
   = 16 + 100 \\
   = 116 \\
 \]
Now this vector is rotated in the XY plane where its x-coordinate gets doubled, as we know the magnitude of the vector upon rotation does not change hence we can say
\[A = A'\][\[A'\]is the new vector]
Therefore we can write
\[
  {A^2} = x_2^2 + y_2^2 \\
  {A^2} = {8^2} + y_2^2 \\
  116 = 64 + y_2^2 \\
  y_2^2 = 116 - 64 \\
  {y_2} = \sqrt {52} \\
   = 7.2m \\
 \]
Hence the new y component \[ = 7.2m\]
Option (B) is correct

Note:On rotation, the magnitude of vector does not change only its direction changes. Vectors whose magnitude is proportional to the amount of rotation and whose direction is perpendicular to the plane of rotation are rotation vectors.