Vectors and Transformations

If \( \vec{a} = 3\hat{i} + 4\hat{j} \) and \( \vec{b} = \hat{i} - 2\hat{j} \), what is \( \vec{a} + \vec{b} \)?

A point \( P(2,3) \) is translated by the vector \( \vec{v} = 5\hat{i} - 2\hat{j} \). What are the coordinates of the image point \( P' \)?

Which of the following transformations is an isometry?

If \( \vec{u} = 2\hat{i} - 3\hat{j} \), find the magnitude of \( 3\vec{u} \).

A point \( Q(4, -1) \) is reflected about the y-axis. What are the coordinates of the reflected point?

The vector \( \vec{p} = \hat{i} + 2\hat{j} \) is rotated 90° anticlockwise about the origin. What is the new vector?

If \( \vec{a} = 5\hat{i} - 2\hat{j} \) and \( \vec{b} = -3\hat{i} + 4\hat{j} \), find the scalar product \( \vec{a} \cdot \vec{b} \).

A triangle with vertices \( A(1,2), B(4,3), C(3,5) \) is dilated with center at origin and scale factor 2. What are the coordinates of \( C' \)?

Vector \( \vec{v} = 6\hat{i} + 8\hat{j} \) is normalized. What is the unit vector in the direction of \( \vec{v} \)?