Vector Algebra

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

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

The vector \( \vec{c} = \hat{i} - 2\hat{j} + 2\hat{k} \) is perpendicular to which of the following vectors?

If \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k} \), find the vector product \( \vec{a} \times \vec{b} \).

If \( \vec{a} \) and \( \vec{b} \) are non-zero vectors such that \( \vec{a} \times \vec{b} = \vec{0} \), then which of the following is true?

Find the scalar triple product \( [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \) if \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k} \), and \( \vec{c} = 7\hat{i} + 8\hat{j} + 9\hat{k} \).

If \( \vec{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), find the unit vector in the direction of \( \vec{a} \).

Vectors \( \vec{a} = 2\hat{i} + \hat{j} - 3\hat{k} \) and \( \vec{b} = \lambda \hat{i} + 3\hat{j} + 4\hat{k} \) are perpendicular. Find \( \lambda \).

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 angle between \( \vec{a} \) and \( \vec{b} \).

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