Vectors in three dimensions – Year 13 A level maths worksheet with answers to download as PDF

\(\mathbf a=3\mathbf i-2\mathbf j+5\mathbf k\) and \(\mathbf b=-\mathbf i+4\mathbf j+2\mathbf k\).

1. Find \(\mathbf a+\mathbf b\). [1 mark]
2. Find \(3\mathbf a-2\mathbf b\). [2 marks]
3. Find \(|\mathbf a|\) and \(|\mathbf a+\mathbf b|\), giving answers to 3 significant figures. [3 marks]

Let \(\mathbf v=2\mathbf i+3\mathbf j+6\mathbf k\).

1. Find \(|\mathbf v|\). [1 mark]
2. Write down the unit vector in the direction of \(\mathbf v\). [1 mark]
3. Find a vector of magnitude 21 in the direction of \(\mathbf v\). [2 marks]
4. Given that \(|4\mathbf i-\lambda\mathbf j+8\mathbf k|=12\), find the possible values of \(\lambda\). [3 marks]

\(A\) is the point \((2,-1,4)\) and \(B\) is the point \((5,3,-8)\).

1. Find \(\overrightarrow{AB}\). [1 mark]
2. Find the distance \(AB\). [2 marks]
3. Find the position vector of the midpoint of \(AB\). [2 marks]
4. The point \(P\) lies on \(AB\) with \(AP:PB=2:3\). Find the coordinates of \(P\). [3 marks]

The points \(A(1,0,2)\), \(B(4,3,8)\) and \(C(6,5,12)\) are given.

1. Find \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). [2 marks]
2. Show that \(A\), \(B\) and \(C\) are collinear. [2 marks]
3. Find the ratio \(AB:BC\). [1 mark]

Find the values of \(p\) and \(q\) for which each pair of vectors is parallel.

1. \(p\mathbf i+q\mathbf j+12\mathbf k\) and \(2\mathbf i-3\mathbf j+4\mathbf k\). [3 marks]
2. \(6\mathbf i+9\mathbf j-3\mathbf k\) and \(4\mathbf i+p\mathbf j+q\mathbf k\). [3 marks]

\(ABCD\) is a parallelogram with \(A(1,2,0)\), \(B(4,3,1)\) and \(C(6,7,5)\).

1. Find the coordinates of \(D\). [3 marks]
2. Show that the diagonals \(AC\) and \(BD\) have the same midpoint. [2 marks]

A drone is launched from a rooftop. At time \(t\) seconds its position vector relative to a fixed origin is \(\mathbf r=(5+2t)\mathbf i+(3-t)\mathbf j+(10+2t)\mathbf k\) metres, where \(\mathbf k\) is vertically upwards.

1. Find the position of the drone at launch and its height above the origin. [2 marks]
2. Show that the drone flies at constant velocity and find its speed. [3 marks]
3. Find its position when \(t=10\). [1 mark]
4. The drone has to stay below a height of 40 m. Find the time at which it reaches this height. [2 marks]

A rectangular classroom is 8 m long, 6 m wide and 3 m high. A projector cable runs in a straight line from a corner of the floor to the opposite corner of the ceiling.

1. Taking one floor corner as the origin, write down the coordinates of the opposite ceiling corner. [1 mark]
2. Find the length of the cable. [2 marks]
3. Find the angle the cable makes with the floor. [3 marks]

Two light aircraft fly at constant velocity. Relative to a control tower at the origin, with \(\mathbf k\) vertically upwards and distances in km, their position vectors at time \(t\) minutes are \(\mathbf r_1=(1+2t)\mathbf i+(2+t)\mathbf j+(3-t)\mathbf k\) and \(\mathbf r_2=(11-3t)\mathbf i+2t\mathbf j+(5-2t)\mathbf k\).

1. Find the speed of each aircraft in km per minute. [3 marks]
2. Show that the aircraft collide, and find the position of the collision. [5 marks]
3. A third aircraft has position \(\mathbf r_3=(11-3t)\mathbf i+2t\mathbf j+(6-2t)\mathbf k\). Show that it does not collide with the first aircraft. [3 marks]

A tent has a rectangular base \(ABCD\) with \(A(0,0,0)\), \(B(4,0,0)\), \(C(4,3,0)\) and \(D(0,3,0)\) in metres. The apex \(E\) is at \((2,1.5,2.5)\), directly above the centre of the base.

1. Find \(\overrightarrow{AE}\) and show that \(AE=\dfrac{5\sqrt2}2\) m. [3 marks]
2. Show that all four sloping edges have the same length. [3 marks]
3. Find the angle between a sloping edge and the base. [2 marks]

A zip wire at an adventure park runs from the top of a tower \(A(0,0,24)\) to a landing point \(B(80,60,0)\), with all distances in metres and \(\mathbf k\) vertically upwards.

1. Find \(\overrightarrow{AB}\) and the length of the wire. [3 marks]
2. Find the angle the wire makes with the horizontal. [3 marks]
3. A safety net is at the point on the wire which is 12 m above the ground. Find its coordinates. [2 marks]

A lamp hangs in equilibrium under three forces \(\mathbf F_1=(4\mathbf i-2\mathbf j+6\mathbf k)\) N, \(\mathbf F_2=(-\mathbf i+5\mathbf j-3\mathbf k)\) N and \(\mathbf F_3\).

1. Write down the condition for equilibrium and hence find \(\mathbf F_3\). [3 marks]
2. Find the magnitude of \(\mathbf F_3\). [2 marks]
3. \(\mathbf F_3\) is removed. The lamp has mass 2 kg. Find the magnitude of its acceleration. [3 marks]

A square pyramid roof has base \(ABCD\) with \(A(0,0,0)\), \(B(6,0,0)\), \(C(6,6,0)\), \(D(0,6,0)\) and apex \(V(3,3,4)\) in metres.

1. Find \(\overrightarrow{VA}\) and the length of the edge \(VA\). [3 marks]
2. Find the angle between the edge \(VA\) and the base. [3 marks]
3. Find the coordinates of the midpoint \(M\) of \(BC\) and the length \(VM\). [3 marks]

A telecommunications company wants to place a relay mast on the straight line joining two towers \(A(2,-3,1)\) and \(B(10,5,9)\) (coordinates in km, with the third coordinate the height above sea level). The mast \(P\) should be three quarters of the way from \(A\) to \(B\).

1. Find the coordinates of \(P\). [3 marks]
2. Find the distance \(AB\). [2 marks]
3. Find the distance from \(P\) to \(B\). [1 mark]
4. Find the height gain per kilometre of horizontal distance along the line. [2 marks]

Three joints of a roof truss are at \(A(2,1,3)\), \(B(5,5,15)\) and \(C(2,1,8)\) with all lengths in metres.

1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) and \(\overrightarrow{BC}\). [3 marks]
2. Find the length of each side of triangle \(ABC\). [3 marks]
3. Use the cosine rule to find angle \(BAC\). [3 marks]
4. Find the area of the triangle and the perpendicular distance from \(B\) to \(AC\). [3 marks]

Two drones fly at constant velocity. Relative to an origin \(O\) with distances in km, their position vectors after \(t\) minutes are \(\mathbf r_1=(1+2t)\mathbf i+(2+t)\mathbf j+(3-t)\mathbf k\) and \(\mathbf r_3=(11-3t)\mathbf i+2t\mathbf j+(6-2t)\mathbf k\).

1. Show that the vector from drone 1 to drone 3 is \((10-5t)\mathbf i+(t-2)\mathbf j+(3-t)\mathbf k\). [2 marks]
2. Show that the square of the distance between the drones is \(d^2=27t^2-110t+113\). [3 marks]
3. Find the value of \(t\) for which the drones are closest together and find the minimum distance. [5 marks]

\(OABC\) is a tetrahedron with \(O(0,0,0)\), \(A(4,0,0)\), \(B(0,6,0)\) and \(C(0,0,8)\). \(M\) is the midpoint of \(AB\) and \(G\) is the point on \(CM\) with \(CG:GM=2:1\).

1. Find the coordinates of \(M\). [1 mark]
2. Find the coordinates of \(G\). [3 marks]
3. Show that \(\mathbf g=\tfrac13(\mathbf a+\mathbf b+\mathbf c)\) and describe the point \(G\). [3 marks]
4. Find the lengths \(OG\) and \(CM\). [3 marks]

A shipping box is a cuboid 8 cm by 6 cm by 5 cm. A thin rod rests inside the box along the longest possible straight line, from a bottom corner to the opposite top corner.

1. Find the length of the longest rod that fits in the box. [3 marks]
2. Find the angle the rod makes with the base. [3 marks]
3. Find the angle the rod makes with the 8 cm edge. [3 marks]
4. A rod of length 11.5 cm is posted into the box. Explain whether it fits. [1 mark]

A weather balloon is tracked from a radio mast at the origin. Its position vector after \(t\) hours is \(\mathbf r=(4-t)\mathbf i+(2+2t)\mathbf j+(6-2t)\mathbf k\) km, with \(\mathbf k\) vertically upwards.

1. Show that the balloon moves with constant velocity and find its speed. [3 marks]
2. Show that the distance \(d\) from the mast satisfies \(d^2=9t^2-24t+56\). [3 marks]
3. Find the least distance between the balloon and the mast, and the time at which it occurs. [4 marks]
4. At what time does the balloon reach the ground (\(z=0\))? Is it still within 8 km of the mast then? [3 marks]

\(ABCD\) is a parallelogram.

1. Prove, using position vectors, that the diagonals \(AC\) and \(BD\) bisect each other. [4 marks]
2. \(A(1,2,3)\), \(B(4,6,3)\), \(C(7,6,9)\). Find \(D\) and confirm your result from part (a) numerically. [3 marks]

\(O\) is the origin, \(B\) is \((2,2,1)\) and \(C\) is \((-2,1,\lambda)\).

1. Show that \(\angle BOC=90^\circ\) when \(\lambda=2\). [3 marks]
2. Find the value of \(\lambda\) for which \(\angle BOC=90^\circ\) using Pythagoras, and verify that \(\lambda=2\) is the only solution. [4 marks]
3. Hence find the area of triangle \(OBC\). [2 marks]