Basic properties of vectors in two and three dimensions: Practice Questions

A vector \( \mathbf{p} \) has magnitude 5 and is in the same direction as the vector \( 3\mathbf{i} - 4\mathbf{j} \). Express \( \mathbf{p} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

The point P divides the line segment AB in the ratio \( 2:3 \). If the position vectors of A and B are \( \mathbf{a} \) and \( \mathbf{b} \) respectively, find the position vector of P.

Given two points A and B with coordinates \( (1, 2, 3) \) and \( (4, -2, 3) \) respectively, calculate the distance between \( A \) and \( B \).

Find the magnitude of the vector \( \mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k} \).

Given a point A with position vector \( \mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} \), find the unit vector in the direction of \( \mathbf{a} \).

Given that the vectors \( \mathbf{u} = \begin{pmatrix} 2 \\ a \\ -4 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} -1 \\ 3 \\ b \end{pmatrix} \) are parallel, find the values of the constants \( a \) and \( b \).

The points A, B, and C have position vectors \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - \mathbf{k} \), \( \mathbf{b} = 3\mathbf{i} + 2\mathbf{k} \), and \( \mathbf{c} = k\mathbf{i} + 4\mathbf{j} - 4\mathbf{k} \) respectively, where \( k \) is a constant.
(i) Find the displacement vector \( \vec{AB} \).
(ii) Find the displacement vector \( \vec{AC} \) in terms of \( k \).
(iii) Given that the points A, B, and C are collinear, determine the value of the constant \( k \).