In the given figure, the vectors u and v are related as by a transformation matrix A. The correct choice of matrix A is
% Vector details
u = (4,3), v = (5,0)
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Official Solution
Correct Option: (3)
Step 1: Analyze the vectors and . The initial vector is . The transformed vector is . Let's calculate their magnitudes (lengths): . . Since the magnitudes are equal, the transformation is a rotation.
Step 2: Determine the angle of rotation. The vector makes an angle with the positive x-axis, where and . The vector lies on the positive x-axis, so its angle is 0. To transform to , we need to rotate by an angle of .
Step 3: Construct the rotation matrix. The general rotation matrix for an angle is . We need to use the angle . We know that and . Substituting these into the rotation matrix formula:
Step 4: Verify the result. Let's multiply the matrix with vector : The result matches, so the matrix is correct.
02
PYQ 2025
medium
electronics-engineeringID: cuet-pg-
For a given vector , the vector is normal to the plane defined by .
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Official Solution
Correct Option: (4)
Step 1: Understand the equation of a plane.
The general vector equation of a plane is given by , where is a vector normal (perpendicular) to the plane, is the position vector of any point on the plane, and is a constant related to the distance of the plane from the origin. Step 2: Analyze the given equation.
The given equation is . Let's write this in vector notation.
Let and .
Then .
The equation is the dot product .
This is the standard form of a plane equation. Step 3: Identify the normal vector.
By comparing the given equation with the general form , we can directly identify the normal vector as the vector .
Given , the column vector is , or .
Therefore, the vector normal to the plane is .
