Angle between two vectors/lines (direct)

4 The position vectors of points \(A\) and \(B\) are \(\left( \begin{array} { r } - 3 \\ 6 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right)\) respectively, relative to an origin \(O\).

1. Calculate angle \(A O B\).
2. The point \(C\) is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. The position vectors of points \(C\) and \(D\) relative to \(O\) are \(3 \mathbf { a }\) and \(2 \mathbf { b }\) respectively. It is given that $$\mathbf { a } = \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { b } = \left( \begin{array} { l } 4 \\ 0 \\ 6 \end{array} \right) .$$

1. Find the unit vector in the direction of \(\overrightarrow { C D }\).
2. The point \(E\) is the mid-point of \(C D\). Find angle \(E O D\).

1 Find the angle between the vectors \(\mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).

2 The points \(O ( 0,0,0 ) , A ( 2,8,2 ) , B ( 5,5,8 )\) and \(C ( 3 , - 3,6 )\) form a parallelogram \(O A B C\). Use a scalar product to find the acute angle between the diagonals of this parallelogram.

6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
Given that

• \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
• \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
  1. find \(\overrightarrow { A C }\)
  2. show that \(\cos A B C = \frac { 9 } { 10 }\)

7. Figure 2 Figure 2 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }\),
show that \(\angle B A C = 105.9 ^ { \circ }\) to one decimal place.

10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

9 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

9 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k} \quad \text{and} \quad \overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + 4\mathbf{k}.$$

1. Use a vector method to find angle \(AOB\). [4]

The point \(C\) is such that \(\overrightarrow{AB} = \overrightarrow{BC}\).

1. Find the unit vector in the direction of \(\overrightarrow{OC}\). [4]
2. Show that triangle \(OAC\) is isosceles. [1]

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.

1. Use a scalar product to find angle \(BOA\). [4]

The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).

1. Find \(\overrightarrow{DC}\). [4]

The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to an origin \(O\), where \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\).

1. Find the length of \(AB\). [3]
2. Use a scalar product to find angle \(OAB\). [3]

Verify that the point \((-1, 6, 5)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 0 \\ 6 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}.$$ Find the acute angle between the lines. [7]

Relative to a fixed origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are $$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$

1. Find the cosine of angle \(ABC\). [4]

The line \(L\) is the angle bisector of angle \(ABC\).

1. Show that an equation of \(L\) is \(\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})\). [4]
2. Show that \(|\overrightarrow{AB}| = |\overrightarrow{AC}|\). [2]

The circle \(S\) lies inside triangle \(ABC\) and each side of the triangle is a tangent to \(S\).

1. Find the position vector of the centre of \(S\). [7]
2. Find the radius of \(S\). [5]

Find, to the nearest degree, the angle between the vectors \(\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ -3 \end{pmatrix}\). [3]

The points \(A\) and \(B\) are at \((2, 3, 5)\) and \((8, 2, 4)\) with respect to the origin \(O\).

1. Find the size of angle \(AOB\). [4]
2. Show that triangle \(AOB\) is isosceles. [3]