Perpendicularity conditions

Three points, \(O\), \(A\) and \(B\), are such that \(\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + p\mathbf{k}\) and \(\overrightarrow{OB} = -7\mathbf{i} + (1 - p)\mathbf{j} + p\mathbf{k}\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [3]
2. The magnitudes of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b^2 = 2a^2\). [2]
3. Find the unit vector in the direction of \(\overrightarrow{AB}\) when \(p = -8\). [3]

With respect to a fixed origin, \(O\), the point \(A\) has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$

1. find the coordinates of the point \(B\). [2]

The point \(C\) has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where \(a\) is a constant. Given that \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{BC}\)

1. find the possible values of \(a\). [4]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$$ and $$\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ -6 \end{pmatrix} + t \begin{pmatrix} 3 \\ a \\ b \end{pmatrix},$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters. Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). [2]

Given also that the two lines intersect,

1. find the values of \(a\) and \(b\), [8]
2. find the coordinates of the point where they intersect. [2]

Write down normal vectors to the planes \(2x + 3y + 4z = 10\) and \(x - 2y + z = 5\). Hence show that these planes are perpendicular to each other. [4]

A vector is given by \(\mathbf{a} = \begin{bmatrix} 2 \\ -1 \\ -3 \end{bmatrix}\) Which vector is not perpendicular to \(\mathbf{a}\)? Circle your answer. \(\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}\) \quad \(\begin{bmatrix} 5 \\ -1 \\ 3 \end{bmatrix}\) \quad \(\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}\) [1 mark]

Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   [6]
2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]

1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]

Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\).

1. Find the point of intersection of \(l_1\) and \(l_2\). [4]
2. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]

The plane \(x + 2y + cz = 4\) is perpendicular to the plane \(2x - cy + 6z = 9\), where \(c\) is a constant. Find the value of \(c\). [3]