Use scalar and vector projections of vectors. scalar projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) = 𝒂 ⋅ 𝒃̂ vector projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) 𝒃̂ = (𝒂 ⋅ 𝒃̂)𝒃̂ = (𝒂⋅𝒃 𝒃⋅𝒃) 𝒃

A loading crane has a cable extending from point P to point Q, represented by the vector **PQ** = {2, −3, 4}. A force vector **F** = {6, −2, −1} acts on the cable. (a) Calculate the scalar projection of **F** on **PQ**. Express your answer in the form \(\frac{a\sqrt{b}}{c}\) where \(a\), \(b\), and \(c\) are positive integers with \(b\) square-free. (2 marks) (b) Hence, determine the vector projection of **F** on **PQ**, expressing your answer in the form {\(p\), \(q\), \(r\)} where \(p\), \(q\), \(r\) are simple fractions. (3 marks)

Calculate the scalar projection of vector OP onto vector OQ.

Two vectors are defined as α = (2, 3) and β = (4, 1). Use the scalar product formula to determine the scalar projection of α onto β.

A surveyor needs to find the perpendicular distance from point $P$ to a fence line passing through the origin. The fence direction is given by vector $\mathbf{b} = \begin{pmatrix} 4 \\ 3 \\ 0 \end{pmatrix}$, and the position of $P$ relative to the origin is $\mathbf{p} = \begin{pmatrix} 6 \\ 8 \\ 0 \end{pmatrix}$. The surveyor first calculates the scalar projection of $\mathbf{p}$ onto $\mathbf{b}$ to determine the position along the fence. Which of the following is the scalar projection of $\mathbf{p}$ on $\mathbf{b}$?

Two vectors \(\vec{p} = [2, 1, -1]\) and \(\vec{q} = [1, 0, 1]\) are drawn from the origin as shown. Calculate the scalar projection of \(\vec{p}\) onto \(\vec{q}\), giving your answer in the form \(\frac{a}{\sqrt{b}}\) where \(a\) and \(b\) are positive integers.

Find the scalar projection of \(\mathbf{u} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}\) onto \(\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}\).

The points T(2, −1, 3) and Q(−4, 2, 1) are joined to the origin O. Calculate the scalar projection of vector TQ onto vector OT.

A force **F** = \(\begin{bmatrix} 6 \\ -3 \\ 2 \end{bmatrix}\) newtons acts on an object that moves along a straight line in the direction of vector **d** = \(\begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}\) metres. (a) Calculate the scalar projection of **F** onto **d**. (1 mark) (b) Hence calculate the vector projection of **F** onto **d**. (2 marks)