Angle between two vectors

4. \begin{figure}[h] \end{figure} Figure 2 shows the points \(A\) and \(B\) with position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\). Given that \(| \mathbf { a } | = 5 , | \mathbf { b } | = 6\) and a.b \(= 20\)

1. find the cosine of angle \(A O B\),
2. find the exact length of \(A B\).
3. Show that the area of triangle \(O A B\) is \(5 \sqrt { 5 }\)

5 \includegraphics[max width=\textwidth, alt={}, center]{b5d85e48-0d5a-4edf-bf58-eba4f8d28d3d-2_425_680_1302_689} In the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\). Given that \(| \mathbf { a } | = 3 , | \mathbf { b } | = 4\) and \(\mathbf { a . b } = 6\), find

1. the angle \(A O B\),
2. \(| \mathbf { a } - \mathbf { b } |\).

1. (i) Two non-zero vectors, \(\mathbf { a }\) and \(\mathbf { b }\), are such that

$$| \mathbf { a } + \mathbf { b } | = | \mathbf { a } | + | \mathbf { b } |$$ Explain, geometrically, the significance of this statement.
(ii) Two different vectors, \(\mathbf { m }\) and \(\mathbf { n }\), are such that \(| \mathbf { m } | = 3\) and \(| \mathbf { m } - \mathbf { n } | = 6\) The angle between vector \(\mathbf { m }\) and vector \(\mathbf { n }\) is \(30 ^ { \circ }\) Find the angle between vector \(\mathbf { m }\) and vector \(\mathbf { m } - \mathbf { n }\), giving your answer, in degrees, to one decimal place.

6 You are given that \(\mathbf { v } = 2 \mathbf { a } + 3 \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are the position vectors \(\mathbf { a } = \binom { 5 } { 3 }\) and \(\mathbf { b } = \binom { - 1 } { 6 }\).

1. Determine the magnitude of \(\mathbf { v }\).
2. Determine the angle between \(\mathbf { v }\) and the vector \(\binom { 1 } { 0 }\).

14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
5. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
6. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
7. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
8. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
9. find the value of \(k\),
10. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
11. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
12. find a cartesian equation for the curve.
13. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
14. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    7. continued
    7. continued

3 The points \(P , Q\) and \(R\) have coordinates \(( - 1,6 ) , ( 2,10 )\) and \(( 11,1 )\) respectively. Find the angle \(P R Q\).

7 A lighting engineer is setting up part of a display inside a large building. The diagram shows a plan view of the area in which he is working. He has two lights, which project narrow beams of light. One is set up at a point 3 metres above the point \(A\) and the beam from this light hits the wall 23 metres above the point \(D\). The other is set up 1 metre above the point \(B\) and the beam from this light hits the wall 29 metres above the point \(C\). \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-10_776_1301_826_392} 7

1. By creating a suitable model, show that the beams of light intersect. 7
2. Find the angle between the two beams of light.
   [0pt] [3 marks]
   7
3. State one way in which the model you created in part (a) could be refined.
   [0pt] [1 mark]

11 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }\) respectively, relative to the origin \(O\). The point \(P\) lies on \(O A\) extended so that \(\overrightarrow { O P } = 3 \overrightarrow { O A }\) and the point \(Q\) lies on \(O B\) extended so that \(\overrightarrow { O Q } = 2 \overrightarrow { O B }\).

1. Find the coordinates of the point of intersection of the lines \(A Q\) and \(B P\).
2. Find the acute angle between the lines \(A Q\) and \(B P\).

Relative to the origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 10 \\ 0 \\ 6 \end{pmatrix}.$$

1. Find angle \(ABC\). [6]

The point \(D\) is such that \(ABCD\) is a parallelogram.

1. Find the position vector of \(D\). [2]

1. Find the angle between the vectors \(\mathbf{3i} - \mathbf{4k}\) and \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\). [4]

The vector \(\overrightarrow{OA}\) has a magnitude of \(15\) units and is in the same direction as the vector \(\mathbf{3i} - \mathbf{4k}\). The vector \(\overrightarrow{OB}\) has a magnitude of \(14\) units and is in the same direction as the vector \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\).

1. Express \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
2. Find the unit vector in the direction of \(\overrightarrow{AB}\). [3]

Relative to the origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 0 \\ -4 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 6 \\ -3 \\ 2 \end{pmatrix}.$$

1. Find the cosine of angle \(AOB\). [3]

The position vector of \(C\) is given by \(\overrightarrow{OC} = \begin{pmatrix} k \\ -2k \\ 2k - 3 \end{pmatrix}\).

1. Given that \(AB\) and \(OC\) have the same length, find the possible values of \(k\). [4]

\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]

The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).

1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]

Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).

1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]

The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf{a} = -3\mathbf{i} + 4\mathbf{j}, \quad \mathbf{b} = 5\mathbf{i} + 8\mathbf{j},$$ respectively.

1. Find the vector \(\overrightarrow{AB}\). [2]
2. 1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
   2. The point \(C\) is such that the vector \(\overrightarrow{OC}\) is in the direction of \(\mathbf{a}\). Given that the length of \(\overrightarrow{OC}\) is 7 units, write down the position vector of \(C\). [1]
3. Calculate the angle \(AOB\). [3]