Examine and use addition and subtraction of vectors in Cartesian form.

Three displacement vectors are recorded during a navigation exercise. A surveyor measures the net displacement of a route by combining the three legs. Refer to the table of vectors below. (a) Determine the resultant vector $\vec{R} = \vec{a} + \vec{b} + \vec{c}$ in Cartesian form. [2] (b) Hence, determine the magnitude of the resultant displacement, correct to 2 decimal places. [1] (c) A corrective displacement vector $\vec{d}$ is applied such that $\vec{a} + \vec{b} + \vec{c} + \vec{d} = \vec{0}$. State the vector $\vec{d}$ in Cartesian form. [1]

A logistics company tracks the displacement of delivery vehicles using position vectors. Two vehicles start from a central depot and travel to different locations. Vehicle A travels with displacement vector \(\vec{a} = \begin{pmatrix} 8 \\ -3 \\ 4 \end{pmatrix}\) km, while Vehicle B travels with displacement vector \(\vec{b} = \begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix}\) km. (a) Determine the vector \(\vec{c} = \vec{a} + \vec{b}\) and interpret what this represents in the context of the problem. (2 marks) (b) Calculate the magnitude of \(\vec{c}\) correct to 2 decimal places. (2 marks)

Three position vectors are defined as **a** = 3**i** + 2**j** − 5**k**, **b** = −**i** + 4**j** + 7**k**, and **c** = 2**i** − 3**j** + **k**. A fourth vector **d** is formed such that **d** = **a** − 2**b** + 3**c**. (a) Determine **d** in Cartesian form. (2 marks) (b) Given that a vector **p** = m**i** + n**j** + q**k** satisfies the equation 2**p** + **d** = **a** + **b**, where m, n, q ∈ ℝ, show that **p** = −3**i** + **j** − 4**k**. (3 marks)

Given the vectors **a** = 3**i** + 2**j** − 5**k** and **b** = −**i** + 4**j** + 2**k**, calculate 2**a** − 3**b**.