Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular.

Three vectors are defined as follows: $\vec{u} = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}$, $\vec{v} = \begin{pmatrix} 6 \\ -9 \\ 12 \end{pmatrix}$, and $\vec{w} = \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}$. (a) Examine whether $\vec{u}$ and $\vec{v}$ are parallel. Justify your conclusion. [2 marks] (b) Determine whether $\vec{u}$ and $\vec{w}$ are perpendicular. Show your working. [2 marks]

Three vectors are defined by their components: $\vec{u} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$, $\vec{v} = \begin{pmatrix} a \\ 6 \\ -2 \end{pmatrix}$, and $\vec{w} = \begin{pmatrix} 4 \\ b \\ 2 \end{pmatrix}$. (a) Find the value of $a$ such that $\vec{u}$ and $\vec{v}$ are parallel. (b) Given that $\vec{u}$ and $\vec{w}$ are perpendicular, determine the value of $b$.

Given vectors a = 2i + j + 2k, b = i - 2j + k, and c = mi + nj + 3k where m, n ∈ ℝ. (a) Show that a and b are perpendicular. (1 mark) (b) Determine the value of m such that c is perpendicular to b. (1 mark) (c) Given that c is also perpendicular to a, determine the value of n. (1 mark) (d) Calculate d = a + 2b and determine whether d is parallel to e = 2i - 3j + 2k. (2 marks)

Given the vectors \(\vec{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 4\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\), which statement correctly describes the relationship between \(\vec{a}\) and \(\vec{b}\)?