4.04g Vector product: a x b perpendicular vector

The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\mathbf{a} \times \mathbf{c} = \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}\) Work out \((\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}) \times (2\mathbf{a})\) [4 marks]

Given that the vectors a and b are perpendicular, prove that \(|(\mathbf{a} + 5\mathbf{b}) \times (\mathbf{a} - 4\mathbf{b})| = k|\mathbf{a}||\mathbf{b}|\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof. [9 marks]

The equations of two intersecting lines are $$\mathbf{r} = \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix}$$ where \(a\) is a constant.

1. Find a vector, \(\mathbf{b}\), which is perpendicular to both lines. [2]
2. Show that \(\mathbf{b} \cdot \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} = \mathbf{b} \cdot \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\). [2]
3. Hence, or otherwise, find the value of \(a\). [2]

Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\). [3]

\(M\) is the point on \(l\) that is closest to \(C\).

1. Find the coordinates of \(M\). [4]
2. Find the exact area of the triangle \(ABC\). [4]

Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]

The line \(L_1\) passes through the points \(A(1, 2, -3)\) and \(B(-2, 1, 0)\).

1. 1. Show that the vector equation of \(L_1\) can be written as $$\mathbf{r} = (1 - 3\lambda)\mathbf{i} + (2 - \lambda)\mathbf{j} + (-3 + 3\lambda)\mathbf{k}.$$
   2. Write down the equation of \(L_1\) in Cartesian form. [4]

The vector equation of the line \(L_2\) is given by \(\mathbf{r} = 2\mathbf{i} - 4\mathbf{j} + \mu(4\mathbf{j} + 7\mathbf{k})\).

1. Show that \(L_1\) and \(L_2\) do not intersect. [5]
2. Find a vector in the direction of the common perpendicular to \(L_1\) and \(L_2\). [5]

The position vector of point \(A\) is \(\mathbf{a} = -9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\). The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf{a}\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). [2] \(l\) is also perpendicular to the vector \(\mathbf{b}\) where \(\mathbf{b} = -2\mathbf{i} + \mathbf{j} + \mathbf{k}\).
2. Find a vector which is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). [1]
3. Write down an equation of \(l\) in vector form. [1] \(P\) is a point on \(l\) such that \(PA = 2OA\).
4. Find angle \(POA\) giving your answer to 3 significant figures. [3] \(C\) is a point whose position vector, \(\mathbf{c}\), is given by \(\mathbf{c} = p\mathbf{a}\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf{r} = \mathbf{c} + \mu\mathbf{b}\). The point with position vector \(9\mathbf{i} + 8\mathbf{j} - 12\mathbf{k}\) lies on \(m\).
5. Find the value of \(p\). [3]

Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.

1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\). [3]

\(M\) is the point on \(l\) that is closest to \(C\).

1. Find the coordinates of \(M\). [4]
2. Find the exact area of the triangle \(ABC\). [4]

The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.

1. Find the value of \(c\) and the value of \(d\). [3]

The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.

1. Find the position vector of \(P\). [6]
2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]

1. Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [2]
2. Find the shortest possible vector of the form \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [5]

1. Vector \(\mathbf{v}\) is perpendicular to both \(\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ p \\ p^2 \end{pmatrix}\) where \(p\) is a real number. Show that it is impossible for \(\mathbf{v}\) to be perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ p-1 \end{pmatrix}\). [6]

The equation of a plane, \(\Pi\), is $$\Pi: \mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}.$$

1. Find a vector which is perpendicular to \(\Pi\). [2]
2. Hence find an equation for \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
3. Find in the form \(\sqrt{q}\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. [2]

The points \(A\), \(B\), \(C\) and \(P\) have coordinates \((a, 0, 0)\), \((0, b, 0)\), \((0, 0, c)\) and \((a, b, c)\) respectively, where \(a\), \(b\) and \(c\) are positive constants. The plane \(\Pi\) contains \(A\), \(B\) and \(C\).

1. 1. Use the scalar triple product to determine
   2. Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\). [2]
2. 1. Determine a vector which is normal to \(\Pi\). [2]
   2. Hence determine, in terms of \(a\), \(b\) and \(c\) only, the distance from \(P\) to \(\Pi\). [3]

Find the volume of tetrahedron OABC, where O is the origin, A = (2, 3, 1), B = (-4, 2, 5) and C = (1, 4, 4). [3]

Relative to an origin \(O\), the points \(P\), \(Q\) and \(R\) have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.

1. Determine \(\mathbf{p} \times \mathbf{q}\). [2]
2. Deduce the value of \(\alpha\) for which
   1. \(OR\) is normal to the plane \(OPQ\), [1]
   2. the volume of tetrahedron \(OPQR\) is 50, [3]
   3. \(R\) lies in the plane \(OPQ\). [2]

Determine the volume of tetrahedron \(OABC\), where \(O\) is the origin and \(A\), \(B\) and \(C\) are, respectively, the points \((2, 3, -2)\), \((2, 0, 4)\) and \((6, 1, 7)\). [3]

With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find

1. a vector perpendicular to the plane \(OAB\), [2]
2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]