1.10b Vectors in 3D: i,j,k notation

A particle \(P\) moves in the \(x\)-\(y\) plane so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation $$\frac{d^2\mathbf{r}}{dt^2} - 4\mathbf{r} = -3e^t\mathbf{j}$$ When \(t = 0\), the particle is at the origin and is moving with velocity \((2i + j)\) ms\(^{-1}\). Find \(\mathbf{r}\) in terms of \(t\). [10]

The points \(P\) and \(Q\) have position vectors \(4i - 6j - 12k\) and \(2i + 4j + 4k\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes \(7\) N, \(3\) N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]

1. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2i - 10j - 16k)\) N. [2]

1. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

Three forces \(\mathbf{F}_1 = (3i - j + k)\) N, \(\mathbf{F}_2 = (2i - k)\) N, and \(\mathbf{F}_3\) act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((i + 2j + k)\) m, the force \(\mathbf{F}_2\) acts through the point with position vector \((i - 2j)\) m and the force \(\mathbf{F}_3\) acts through the point with position vector \((i + j + k)\) m. Given that the system \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) reduces to a couple \(\mathbf{G}\),

1. find \(\mathbf{G}\). [6]

The line of action of \(\mathbf{F}_3\) is changed so that the system \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) now reduces to a couple \((6i + 8j + 2k)\) N m.

1. Find an equation of the new line of action of \(\mathbf{F}_3\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors. [5]

A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\). [9]

A force system consists of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a rigid body. \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j})\) N and acts at the point with position vector \((-\mathbf{i} + 4\mathbf{j})\) m. \(\mathbf{F}_2 = (-\mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((2\mathbf{i} + \mathbf{j} + \mathbf{k})\) m. \(\mathbf{F}_3 = (3\mathbf{i} - \mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) m. It is given that this system can be reduced to a single force \(\mathbf{R}\).

1. Find \(\mathbf{R}\). [2]
2. Find a vector equation of the line of action of \(\mathbf{R}\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [10]

Two forces \(\mathbf{F}_1 = (3\mathbf{i} + \mathbf{k})\) N and \(\mathbf{F}_2 = (4\mathbf{i} + \mathbf{j} - \mathbf{k})\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3\mathbf{i} + 2\mathbf{k})\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2\mathbf{i} - \mathbf{k})\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

The points \(O\) and \(A\) have position vectors \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 0 \\ 8 \end{pmatrix}\) respectively. The point \(P\) is such that \(\overrightarrow{OP} = k\overrightarrow{OA}\), where \(k\) is a non-zero constant.

1. Find, in terms of \(k\), the length of \(OP\). [1] Point \(B\) has position vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and angle \(OPB\) is a right angle.
2. Determine the value of \(k\). [4]

A particle, \(P\), is moving with constant velocity \(8\mathbf{i} - 12\mathbf{j}\) A second particle, \(Q\), is moving with constant velocity \(a\mathbf{i} + 9\mathbf{j}\) \(Q\) travels in a direction which is parallel to the motion of \(P\). Find \(a\). Circle your answer. \(-6\) \quad \(-5\) \quad \(5\) \quad \(6\) [1 mark]

Two particles \(P\) and \(Q\) are moving in separate straight lines across a smooth horizontal surface. \(P\) moves with constant velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) \(Q\) moves from position vector \((5\mathbf{i} - 7\mathbf{j})\) metres to position vector \((14\mathbf{i} + 5\mathbf{j})\) metres during a 3 second period.

1. Show that \(P\) and \(Q\) move along parallel lines. [3 marks]
2. Stevie says Q is also moving with a constant velocity of \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) Explain why Stevie may be incorrect. [1 mark]
3. A third particle \(R\) is moving with a constant speed of 4 m s\(^{-1}\), in a straight line, across the same surface. \(P\) and \(R\) move along lines that intersect at a fixed point \(X\) It is given that: • \(P\) passes through \(X\) exactly 2 seconds after \(R\) passes through \(X\) • \(P\) and \(R\) are exactly 13 metres apart 3 seconds after \(R\) passes through \(X\) Show that \(P\) and \(R\) move along perpendicular lines. [5 marks]

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]

A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the magnitude of the acceleration of the particle. [3]
2. Given that at time \(t = 0\) seconds, the position vector of the particle is \(2\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}\) and it is moving with velocity \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), find the position vector of the particle when \(t = 2\) seconds. [3]

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]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. Distance is measured in metres and time in seconds. A ship of mass 100 000 kg is being towed by two tug boats. • The cables attaching each tug to the ship are horizontal. • One tug produces a force of \((350\mathbf{i} + 400\mathbf{j})\) N. • The other tug produces a force of \((250\mathbf{i} - 400\mathbf{j})\) N. • The total resistance to motion is 200 N. • At the instant when the tugs begin to tow the ship, it is moving east at a speed of 1.5 m s\(^{-1}\).

1. Explain why the ship continues to move directly east. [2]
2. Find the acceleration of the ship. [2]
3. Find the time which the ship takes to move 400 m while it is being towed. Find its speed after moving that distance. [6]

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]

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]

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 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]