Vector Algebra

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

If \( \vec{a} = 5\hat{i} - 3\hat{j} \), what is the magnitude of \( \vec{a} \)?

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

Two vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular. If \( \vec{a} = 3\hat{i} + 4\hat{j} \), which of the following can be \( \vec{b} \)?

If \( \vec{a} = \hat{i} + 2\hat{j} \) and \( \vec{b} = 3\hat{i} + k\hat{j} \) are parallel, find \( k \).

What is the vector \( \vec{c} \) if \( \vec{c} = 2\vec{a} - 3\vec{b} \), where \( \vec{a} = \hat{i} - \hat{j} \) and \( \vec{b} = 2\hat{i} + 3\hat{j} \)?

If \( \vec{a} = 4\hat{i} + 3\hat{j} \), find the unit vector in the direction of \( \vec{a} \).

If \( \vec{a} = 2\hat{i} + 3\hat{j} \) and \( \vec{b} = 5\hat{i} + 7\hat{j} \), what is the angle between \( \vec{a} \) and \( \vec{b} \)?

If \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 4\hat{i} - \hat{j} + 2\hat{k} \), find \( \vec{a} \times \vec{b} \).

If \( \vec{a} = 3\hat{i} + 4\hat{j} \) and \( \vec{b} = k\hat{i} + 6\hat{j} \), find \( k \) such that \( \vec{a} \) and \( \vec{b} \) are orthogonal.