1.10c Magnitude and direction: of vectors

A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]

A particle P moves so that its position vector \(\mathbf{r}\) at time \(t\) is given by $$\mathbf{r} = (5 + 20t)\mathbf{i} + (95 + 10t - 5t^2)\mathbf{j}.$$

1. Determine the initial velocity of P. [3] At time \(t = T\), P is moving in a direction perpendicular to its initial direction of motion.
2. Determine the value of \(T\). [3]
3. Determine the distance of P from its initial position at time \(T\). [4]

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are defined by \(\mathbf{a} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\).

1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
2. Determine the angle \(\mathbf{b}\) makes with the \(x\)-axis. [2]
3. The vector \(\mu\mathbf{a} + \mathbf{b}\) is parallel to \(4\mathbf{i} - 5\mathbf{j}\).
   1. Find the vector \(\mu\mathbf{a} + \mathbf{b}\) in terms of \(\mu\), \(\mathbf{i}\) and \(\mathbf{j}\). [1]
   2. Determine the value of \(\mu\). [4]

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]

Three forces \(\mathbf{L}\), \(\mathbf{M}\) and \(\mathbf{N}\) are given by $$\mathbf{L} = 2\mathbf{i} + 5\mathbf{j},$$ $$\mathbf{M} = 3\mathbf{i} - 22\mathbf{j},$$ $$\mathbf{N} = 4\mathbf{i} - 23\mathbf{j}.$$ Find the magnitude and direction of the resultant of the three forces. [6]

A ship \(S\) is moving with constant velocity \((4\mathbf{i} - 7\mathbf{j})\text{ms}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Find the speed and direction of \(S\), giving the direction as a three-figure bearing, correct to the nearest degree. [4]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) acting on an object are such that $$\mathbf{F} = \mathbf{i} - 8\mathbf{j},$$ $$\mathbf{G} = 3\mathbf{i} + 11\mathbf{j}.$$ The object has a mass of 3 kg. Calculate the magnitude and direction of the acceleration of the object. [7]

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]

Given that the point \(A\) has position vector \(3\mathbf{i} - 7\mathbf{j}\) and the point \(B\) has position vector \(8\mathbf{i} + 3\mathbf{j}\),

1. find the vector \(\overrightarrow{AB}\) [2]
2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]

Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.

1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]

Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(-2\mathbf{i} + 3\mathbf{j}\),
• the point \(B\) has position vector \(3\mathbf{i} + p\mathbf{j}\), where \(p\) is constant,

Given that \(|\overrightarrow{AB}| = 5\sqrt{2}\), find the possible values for \(p\). [3]

A particle \(P\) is acted upon by three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) given by \(\mathbf{F}_1 = (6\mathbf{i} - 4\mathbf{j}) \text{ N}\), \(\mathbf{F}_2 = (-3\mathbf{i} + 9\mathbf{j}) \text{ N}\) and \(\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j}) \text{ N}\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,

1. find the value of \(a\) and the value of \(b\). [2]

The force \(\mathbf{F}_3\) is now removed. The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\).

1. Find the magnitude of \(\mathbf{R}\). [3]
2. Find the angle, to \(0.1°\), that \(\mathbf{R}\) makes with \(\mathbf{i}\). [2]

Relative to a fixed origin \(O\), • the point \(A\) has position vector \(\mathbf{5i + 3j - 2k}\) • the point \(B\) has position vector \(\mathbf{7i + j + 2k}\) • the point \(C\) has position vector \(\mathbf{4i + 8j - 3k}\)

1. Find \(|\overrightarrow{AB}|\) giving your answer as a simplified surd. [2]

Given that \(ABCD\) is a parallelogram,

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

The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC:CE = 2:1\)

1. Find the coordinates of the point \(E\). [2]

\includegraphics{figure_5} Figure 1 shows a sketch of a triangle \(ABC\). Given \(\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{BC} = \mathbf{i} - 9\mathbf{j} + 3\mathbf{k}\), show that \(\angle BAC = 105.9°\) to one decimal place. [5]

Relative to a fixed origin \(O\), • the point \(A\) has position vector \(5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) • the point \(B\) has position vector \(7\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) • the point \(C\) has position vector \(4\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\)

1. Find \(|\vec{AB}|\) giving your answer as a simplified surd. [2] Given that \(ABCD\) is a parallelogram,
2. find the position vector of the point \(D\). [2] The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC : CE = 2 : 1\)
3. Find the coordinates of the point \(E\). [2]

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(4\mathbf{i} + 2\mathbf{j}\), \(3\mathbf{i} + 4\mathbf{j}\) and \(-\mathbf{i} + 12\mathbf{j}\), respectively.

1. Find the magnitude of the vector \(\overrightarrow{OC}\) [2]
2. Find the angle that the vector \(\overrightarrow{OB}\) makes with the vector \(\mathbf{j}\) to the nearest degree [2]
3. Show that the points \(A\), \(B\) and \(C\) are collinear [3]

Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are given by $$\mathbf{F}_1 = 13\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}, \quad \mathbf{F}_2 = -2\mathbf{i} + 6\mathbf{j} + \mathbf{k},$$ in which the units of the components are newtons. A third force, \(\mathbf{F}_3\), of magnitude 6 N acts parallel to the vector \(2\mathbf{i} - 2\mathbf{j} + \mathbf{k}\).

1. Find the two possible resultants of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), and show that they have the same magnitude. [5]

A particle, \(P\), of mass 2 kg is initially at rest at the origin. The only forces acting on \(P\) are \(\mathbf{F}_1\) and \(\mathbf{F}_2\).

1. Find the magnitude of the acceleration of \(P\). [3]
2. Find the time taken for \(P\) to travel 60 m. [2]

A particle is being held in equilibrium by the following set of forces (in newtons). $$\mathbf{F}_1 = 5\mathbf{i} - 8\mathbf{j}, \quad \mathbf{F}_2 = -3\mathbf{i} - 4\mathbf{j}, \quad \mathbf{F}_3 = 6\mathbf{i} + 6\mathbf{j} \quad \text{and} \quad \mathbf{F}_4.$$

1. Find \(\mathbf{F}_4\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [2]
2. Hence find the magnitude and direction of \(\mathbf{F}_4\). [4]

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]

The planes \(\Pi_1\) and \(\Pi_2\) are both perpendicular to \(\mathbf{n}\), where \(\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). The points \(A(0, -9, 13)\) and \(B(8, 7, -3)\) lie in \(\Pi_1\) and \(\Pi_2\) respectively.

1. Find the equations of \(\Pi_1\) and \(\Pi_2\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\) and show that \(\overrightarrow{AB}\) is parallel to \(\mathbf{n}\). [4]
2. Calculate the perpendicular distance between \(\Pi_1\) and \(\Pi_2\). [2]
3. Write down two vectors which are perpendicular to \(\mathbf{n}\) and hence find, in the form $$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$ an equation for the plane \(\Pi_3\) which is parallel to \(\Pi_1\) and \(\Pi_2\) and exactly half-way between them. [4]
4. The locus of all points \(P\) such that \(AP = BP = 12\sqrt{2}\) is denoted by \(L\).
   1. Give a full geometrical description of \(L\). [4]
   2. Using the result of part (iii), or otherwise, find a point on \(L\) which has integer coordinates. [4]

\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are \(-5\) N and 7 N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]

\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are -5 N and 7 N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]

The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$

1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\).
   [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron. [3]

The line \(L\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$ The point \(P\) has position vector \(\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}\). The point \(P'\) is the reflection of \(P\) in \(L\).

1. Find the position vector of \(P'\). [6]
2. Show that the point \(A\) with position vector \(\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}\) lies on \(L\). [1]
3. Show that angle \(PAP' = 120°\). [3]

% Figure 3 shows kite APBP' with angle at A = 120° \includegraphics{figure_3} Figure 3 The point \(B\) lies on \(L\) and \(APBP'\) forms a kite as shown in Figure 3. The area of the kite is \(50\sqrt{3}\)

1. Find the position vector of the point \(B\). [5]
2. Show that angle \(BPA = 90°\). [2]

The circle \(C\) passes through the points \(A\), \(P\), \(P'\) and \(B\).

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

[Total 19 marks]

The lines \(L_1\) and \(L_2\) have vector equations $$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$ $$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$

1. Show that \(L_1\) and \(L_2\) are perpendicular. [2]
2. Show that \(L_1\) and \(L_2\) are skew lines. [3] The point \(A\) with position vector \(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) lies on \(L_2\) and the point \(X\) lies on \(L_1\) such that \(\overrightarrow{AX}\) is perpendicular to \(L_1\)
3. Find the position vector of \(X\). [5]
4. Find \(|\overrightarrow{AX}|\) [2] The point \(B\) (distinct from \(A\)) also lies on \(L_2\) and \(|\overrightarrow{BX}| = |\overrightarrow{AX}|\)
5. Find the position vector of \(B\). [5]
6. Find the cosine of angle \(AXB\). [2]