Vectors

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

Find the magnitude of vector \( \vec{v} = 5\hat{i} - 12\hat{j} \).

If \( \vec{a} = 2\hat{i} + 3\hat{j} \) and \( \vec{b} = \lambda \hat{i} + 6\hat{j} \), for what value of \( \lambda \) are \( \vec{a} \) and \( \vec{b} \) perpendicular?

Calculate the dot product of \( \vec{p} = 4\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{q} = -\hat{i} + 3\hat{j} + \hat{k} \).

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 \( \vec{b} = 4\hat{i} + 5\hat{j} + 6\hat{k} \), find \( \vec{a} \times \vec{b} \).

Three points in space are \(A(1,2,3)\), \(B(4,0,1)\), and \(C(2,3,5)\). Find the vector \( \vec{AB} + \vec{BC} \).

If \( \vec{u} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{v} \) is perpendicular to \( \vec{u} \) and has magnitude 3, which of the following could be \( \vec{v} \)?

Given \( \vec{a} = \hat{i} + 2\hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} + 3\hat{j} + \lambda \hat{k} \), find \( \lambda \) such that \( \vec{a} \times \vec{b} = \vec{0} \).

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