Vector Algebra

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

The magnitude of the vector \( \vec{v} = 5\hat{i} - 12\hat{j} \) is:

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

Two vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular if and only if:

Find the vector projection of \( \vec{a} = 3\hat{i} + 4\hat{j} \) on \( \vec{b} = \hat{i} + 2\hat{j} \).

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

If \( \vec{a} \) and \( \vec{b} \) are non-zero vectors such that \( \vec{a} \times \vec{b} = \vec{0} \), then:

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

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

Given \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k} \), find the area of the parallelogram formed by \( \vec{a} \) and \( \vec{b} \).