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Magnitude and direction of a vector

40 questions

LessonNot started
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question, i\mathbf{i} is a unit vector due east and j\mathbf{j} is a unit vector due north. Position vectors are given relative to a fixed origin OO.]

At time tt seconds, t≥1t \ge 1, the position vector of a particle PP is r\mathbf{r} metres, where

r=(2t3−8t2+c) i+(t3−2t2) j\mathbf{r} = (2t^3 - 8t^2 + c)\,\mathbf{i} + (t^3 - 2t^2)\,\mathbf{j}

and cc is a constant.

When t=3t = 3, the bearing of PP from OO is 045∘045^\circ
(a) Show that c=27c = 27

(3 marks)
(b) Find the speed of PP when t=3t = 3

(4 marks)

When t=Tt = T, PP is accelerating in the direction of (2i+5j)(2\mathbf{i} + 5\mathbf{j}).
(c) Find the value of TT.

(4 marks)
●●●●●Level 511 marksStart
The straight line ll passes through the points AA and BB.

Relative to a fixed origin OO
  • the point AA has position vector −2i+4j+k-2\mathbf{i} + 4\mathbf{j} + \mathbf{k}
  • the point BB has position vector 4i+j+4k4\mathbf{i} + \mathbf{j} + 4\mathbf{k}
(a) Find AB→\overrightarrow{AB}

(1 mark)

Given that a point PP lies on ll such that

∣AP→∣=2∣BP→∣\left|\overrightarrow{AP}\right| = 2\left|\overrightarrow{BP}\right|
(b) find the possible position vectors of PP.

(4 marks)
(c) State, giving a reason, which of the two possible positions of PP lies between AA and BB.

(1 mark)
●●●●●Level 56 marksStart
Relative to a fixed origin OO
  • the point AA has position vector i+4j+8k\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}
  • the point BB has position vector 7i+qj−4k7\mathbf{i} + q\mathbf{j} - 4\mathbf{k}
where qq is a constant.
(a) Show that ∣OA→∣=9\left|\overrightarrow{OA}\right| = 9

(1 mark)

Given that ∣OB→∣=∣OA→∣\left|\overrightarrow{OB}\right| = \left|\overrightarrow{OA}\right|
(b) find the possible values of qq.

(2 marks)

For the positive value of qq, the point CC is such that OACBOACB is a parallelogram.
(c) Find the exact area of OACBOACB.

(3 marks)
●●●●●Level 56 marksStart
In this question the unit vectors i\mathbf{i} and j\mathbf{j} are in the directions east and north respectively.

A particle PP of mass 1.21.2 kg is moving on a smooth horizontal surface under the action of a single horizontal force F\mathbf{F} N. At time tt seconds, where t≥0t \geq 0, the velocity v m s−1\mathbf{v}\ \text{m s}^{-1} of PP, relative to a fixed origin OO, is given by

v=(t2−8t+20)i+(t−6)j.\mathbf{v} = (t^2 - 8t + 20)\mathbf{i} + (t - 6)\mathbf{j}.
(a) Show that PP is never stationary.

(2 marks)
(b) Find, in terms of i\mathbf{i} and j\mathbf{j}, the acceleration of PP at time tt.

(1 mark)
(c) Determine the magnitude of F\mathbf{F} when PP is moving in the direction of the vector 5i−j5\mathbf{i} - \mathbf{j}. Give your answer correct to 3 significant figures.

(5 marks)

When t=3t = 3, PP is at the point with position vector (−9i+1.5j)(-9\mathbf{i} + 1.5\mathbf{j}) m.
(d) Determine the bearing of PP from OO at time t=6t = 6.

(5 marks)
●●●●●Level 513 marksStart
The points AA, BB and CC have coordinates (1,0,2)(1, 0, 2), (5,8,−2)(5, 8, -2) and (7,6,k)(7, 6, k) respectively, where kk is a constant.
(a) Given that angle ACBACB is a right angle, find the possible values of kk. (3 marks)
(b) Given also that k>0k > 0, find the exact shortest distance from CC to the line ABAB. (3 marks)
(c) The point DD is the point on ABAB that is closest to CC. Find the ratio AD:DBAD : DB in its simplest form. (2 marks)
●●●●●Level 58 marksStart
Relative to a fixed origin OO, the points AA and BB have position vectors 3j−3k3\mathbf{j} - 3\mathbf{k} and 6i+3k6\mathbf{i} + 3\mathbf{k} respectively.

The point PP lies on the line segment ABAB and is such that OPOP is perpendicular to ABAB.
(a) Find the position vector of PP. (5 marks)
(b) Hence find the exact area of triangle OABOAB. (2 marks)
●●●●●Level 57 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=−i+j+4k\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}, b=5i+4j−2k\qquad \mathbf{b} = 5\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}, c=5i+7j+4k\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}.
(a) Show that angle BAC=π4BAC = \dfrac{\pi}{4}. (4 marks)

The point PP lies on the line segment ABAB and is such that angle APC=π2APC = \dfrac{\pi}{2}.
(b) Find the position vector of PP. (3 marks)
(c) Find the area of triangle ABCABC. (2 marks)
●●●●●Level 59 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=i+2k,b=4i+3j+5kandc=7i+6j−k.\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.
(a) Show that angle ABCABC is a right angle. (3 marks)
(b) The point DD lies on ACAC and is such that BDBD is perpendicular to ACAC. Determine the position vector of DD. (5 marks)
(c) Find the exact length of BDBD. (1 mark)
●●●●●Level 59 marksStart
[In this question i\mathbf{i} and j\mathbf{j} are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin OO.]

A patrol boat AA and a lifeboat BB are each moving with constant velocity.
  • the velocity of AA is 36 km h−136\ \text{km h}^{-1} due east
  • the velocity of BB is 29 km h−129\ \text{km h}^{-1} on a bearing of α\alpha, where tan⁡α=2120\tan\alpha = \dfrac{21}{20}
The boats are modelled as particles.
(a) Find, in terms of i\mathbf{i} and j\mathbf{j}, the velocity of BB in km h−1\text{km h}^{-1}

(2 marks)

At 13:00
  • the position vector of AA is (−4i+12j)(-4\mathbf{i} + 12\mathbf{j}) km
  • the position vector of BB is (5i+5j)(5\mathbf{i} + 5\mathbf{j}) km
At time tt hours after 13:00
  • the position vector of AA is r\mathbf{r} km, where r=(−4+36t)i+12j\mathbf{r} = (-4 + 36t)\mathbf{i} + 12\mathbf{j}
  • the position vector of BB is s\mathbf{s} km
(b) Find an expression for s\mathbf{s} in terms of tt, i\mathbf{i} and j\mathbf{j}.

(2 marks)
(c) Show that at time tt hours after 13:00, AB→=[(9−15t)i+(20t−7)j]\overrightarrow{AB} = [(9 - 15t)\mathbf{i} + (20t - 7)\mathbf{j}] km

(2 marks)
(d) Show that the boats will never collide.

(3 marks)
(e) Find the distance between the boats when BB is north-east of AA.

(4 marks)
●●●●●Level 513 marksStart
[In this question i\mathbf{i} and j\mathbf{j} are horizontal unit vectors.]

Two drones, AA and BB, are flying at the same constant height. Each drone is moving with constant acceleration and the drones are modelled as particles.

At time tt seconds, drone AA has velocity vA m s−1\mathbf{v}_A\ \text{m s}^{-1}, where

vA=4i+2j+(−i+3j)t\mathbf{v}_A = 4\mathbf{i} + 2\mathbf{j} + (-\mathbf{i} + 3\mathbf{j})t
(a) Find the magnitude of the acceleration of AA.

(2 marks)

When t=1t = 1, the velocity of BB is (5i+2j) m s−1(5\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-1}

When t=4t = 4, the velocity of BB is (−i+5j) m s−1(-\mathbf{i} + 5\mathbf{j})\ \text{m s}^{-1}
(b) Find the acceleration of BB, giving your answer in terms of i\mathbf{i} and j\mathbf{j}.

(2 marks)
(c) Find the velocity of BB at time t=0t = 0, giving your answer in terms of i\mathbf{i} and j\mathbf{j}.

(2 marks)

At time T1T_1 seconds, where T1>0T_1 > 0, both drones are moving with the same speed.
(d) Find the exact value of T1T_1

(4 marks)

At time T2T_2 seconds, where T2>0T_2 > 0, both drones are moving in the same direction.
(e) Show that 5T22+pT2+q=05T_2^2 + pT_2 + q = 0, where pp and qq are integers to be found.

(3 marks)
●●●●●Level 513 marksStart
Triangle PQRPQR has PQ⃗=2i+2j−k\vec{PQ} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k} and QR⃗=2i−j+2k\vec{QR} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}. Find the size of angle PQRPQR.
●●●●●Level 53 marksStart

Position vectors

44 questions

LessonNot started
The straight line ll passes through the points AA and BB.

Relative to a fixed origin OO
  • the point AA has position vector −2i+4j+k-2\mathbf{i} + 4\mathbf{j} + \mathbf{k}
  • the point BB has position vector 4i+j+4k4\mathbf{i} + \mathbf{j} + 4\mathbf{k}
(a) Find AB→\overrightarrow{AB}

(1 mark)

Given that a point PP lies on ll such that

∣AP→∣=2∣BP→∣\left|\overrightarrow{AP}\right| = 2\left|\overrightarrow{BP}\right|
(b) find the possible position vectors of PP.

(4 marks)
(c) State, giving a reason, which of the two possible positions of PP lies between AA and BB.

(1 mark)
●●●●●Level 56 marksStart
Relative to a fixed origin OO
  • the point AA has position vector i+4j+8k\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}
  • the point BB has position vector 7i+qj−4k7\mathbf{i} + q\mathbf{j} - 4\mathbf{k}
where qq is a constant.
(a) Show that ∣OA→∣=9\left|\overrightarrow{OA}\right| = 9

(1 mark)

Given that ∣OB→∣=∣OA→∣\left|\overrightarrow{OB}\right| = \left|\overrightarrow{OA}\right|
(b) find the possible values of qq.

(2 marks)

For the positive value of qq, the point CC is such that OACBOACB is a parallelogram.
(c) Find the exact area of OACBOACB.

(3 marks)
●●●●●Level 56 marksStart
Relative to a fixed origin OO, the line ll has equation

r=(−i+3j+4k)+λ(2i+j−2k)\mathbf{r} = (-\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})

where λ\lambda is a scalar parameter.

The point AA and the point BB lie on line ll.

Given that
  • AA has coordinates (1,a,2)(1, a, 2)
  • BB has coordinates (b,6,−2)(b, 6, -2)
(a) find the value of the constant aa and the value of the constant bb.

(2 marks)
(b) Hence show that AB→=4i+2j−4k\overrightarrow{AB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}

(2 marks)

The point CC has coordinates (3,9,5)(3, 9, 5).
(c) Find the size of angle BACBAC, giving your answer in degrees to one decimal place.

(4 marks)

The point DD lies on ll such that the area of triangle ACDACD is three times the area of triangle ABCABC.
(d) Find the coordinates of the two possible positions of DD.

(4 marks)
●●●●●Level 512 marksStart
The points AA, BB and CC have coordinates (1,0,2)(1, 0, 2), (5,8,−2)(5, 8, -2) and (7,6,k)(7, 6, k) respectively, where kk is a constant.
(a) Given that angle ACBACB is a right angle, find the possible values of kk. (3 marks)
(b) Given also that k>0k > 0, find the exact shortest distance from CC to the line ABAB. (3 marks)
(c) The point DD is the point on ABAB that is closest to CC. Find the ratio AD:DBAD : DB in its simplest form. (2 marks)
●●●●●Level 58 marksStart
Relative to a fixed origin OO, the points AA and BB have position vectors 3j−3k3\mathbf{j} - 3\mathbf{k} and 6i+3k6\mathbf{i} + 3\mathbf{k} respectively.

The point PP lies on the line segment ABAB and is such that OPOP is perpendicular to ABAB.
(a) Find the position vector of PP. (5 marks)
(b) Hence find the exact area of triangle OABOAB. (2 marks)
●●●●●Level 57 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=−i+j+4k\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}, b=5i+4j−2k\qquad \mathbf{b} = 5\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}, c=5i+7j+4k\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}.
(a) Show that angle BAC=π4BAC = \dfrac{\pi}{4}. (4 marks)

The point PP lies on the line segment ABAB and is such that angle APC=π2APC = \dfrac{\pi}{2}.
(b) Find the position vector of PP. (3 marks)
(c) Find the area of triangle ABCABC. (2 marks)
●●●●●Level 59 marksStart
Figure 2 shows the triangle OABOAB, where OA→=a\overrightarrow{OA} = \mathbf{a} and OB→=b\overrightarrow{OB} = \mathbf{b}.

The point MM is the midpoint of OBOB. The point NN lies on ABAB with AN:NB=1:3AN : NB = 1 : 3. The lines ONON and AMAM intersect at the point XX.OABMNXabFigure 2 (not accurately drawn)

Given that OX→=λ ON→\overrightarrow{OX} = \lambda\,\overrightarrow{ON} and AX→=μ AM→\overrightarrow{AX} = \mu\,\overrightarrow{AM}, where λ\lambda and μ\mu are scalars, find the value of λ\lambda and the value of μ\mu. (7 marks)
●●●●●Level 57 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=i+2k,b=4i+3j+5kandc=7i+6j−k.\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.
(a) Show that angle ABCABC is a right angle. (3 marks)
(b) The point DD lies on ACAC and is such that BDBD is perpendicular to ACAC. Determine the position vector of DD. (5 marks)
(c) Find the exact length of BDBD. (1 mark)
●●●●●Level 59 marksStart
[In this question i\mathbf{i} and j\mathbf{j} are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin OO.]

A patrol boat AA and a lifeboat BB are each moving with constant velocity.
  • the velocity of AA is 36 km h−136\ \text{km h}^{-1} due east
  • the velocity of BB is 29 km h−129\ \text{km h}^{-1} on a bearing of α\alpha, where tan⁡α=2120\tan\alpha = \dfrac{21}{20}
The boats are modelled as particles.
(a) Find, in terms of i\mathbf{i} and j\mathbf{j}, the velocity of BB in km h−1\text{km h}^{-1}

(2 marks)

At 13:00
  • the position vector of AA is (−4i+12j)(-4\mathbf{i} + 12\mathbf{j}) km
  • the position vector of BB is (5i+5j)(5\mathbf{i} + 5\mathbf{j}) km
At time tt hours after 13:00
  • the position vector of AA is r\mathbf{r} km, where r=(−4+36t)i+12j\mathbf{r} = (-4 + 36t)\mathbf{i} + 12\mathbf{j}
  • the position vector of BB is s\mathbf{s} km
(b) Find an expression for s\mathbf{s} in terms of tt, i\mathbf{i} and j\mathbf{j}.

(2 marks)
(c) Show that at time tt hours after 13:00, AB→=[(9−15t)i+(20t−7)j]\overrightarrow{AB} = [(9 - 15t)\mathbf{i} + (20t - 7)\mathbf{j}] km

(2 marks)
(d) Show that the boats will never collide.

(3 marks)
(e) Find the distance between the boats when BB is north-east of AA.

(4 marks)
●●●●●Level 513 marksStart

Solving geometric problems with vectors

29 questions

LessonNot started
The straight line ll passes through the points AA and BB.

Relative to a fixed origin OO
  • the point AA has position vector −2i+4j+k-2\mathbf{i} + 4\mathbf{j} + \mathbf{k}
  • the point BB has position vector 4i+j+4k4\mathbf{i} + \mathbf{j} + 4\mathbf{k}
(a) Find AB→\overrightarrow{AB}

(1 mark)

Given that a point PP lies on ll such that

∣AP→∣=2∣BP→∣\left|\overrightarrow{AP}\right| = 2\left|\overrightarrow{BP}\right|
(b) find the possible position vectors of PP.

(4 marks)
(c) State, giving a reason, which of the two possible positions of PP lies between AA and BB.

(1 mark)
●●●●●Level 56 marksStart
Relative to a fixed origin OO
  • the point AA has position vector i+4j+8k\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}
  • the point BB has position vector 7i+qj−4k7\mathbf{i} + q\mathbf{j} - 4\mathbf{k}
where qq is a constant.
(a) Show that ∣OA→∣=9\left|\overrightarrow{OA}\right| = 9

(1 mark)

Given that ∣OB→∣=∣OA→∣\left|\overrightarrow{OB}\right| = \left|\overrightarrow{OA}\right|
(b) find the possible values of qq.

(2 marks)

For the positive value of qq, the point CC is such that OACBOACB is a parallelogram.
(c) Find the exact area of OACBOACB.

(3 marks)
●●●●●Level 56 marksStart
Relative to a fixed origin OO, the lines l1l_1 and l2l_2 are given by the equations

l1: r=(−3i+pj−2k)+λ(4i−3j+3k)l_1:\ \mathbf{r} = (-3\mathbf{i} + p\mathbf{j} - 2\mathbf{k}) + \lambda(4\mathbf{i} - 3\mathbf{j} + 3\mathbf{k})

l2: r=(−i−6j−5k)+μ(2i+j+3k)l_2:\ \mathbf{r} = (-\mathbf{i} - 6\mathbf{j} - 5\mathbf{k}) + \mu(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})

where λ\lambda and μ\mu are scalar parameters and pp is a constant.

Given that l1l_1 and l2l_2 intersect,
(a) show that p=3p = 3

(4 marks)
(b) find the position vector of the point of intersection.

(2 marks)
(c) Find the acute angle between l1l_1 and l2l_2. Give your answer in degrees to one decimal place.

(3 marks)

The point AA lies on l1l_1 with parameter λ=1\lambda = 1

The point BB lies on l2l_2 with AB→\overrightarrow{AB} perpendicular to l2l_2
(d) Find the coordinates of BB

(5 marks)
●●●●●Level 514 marksStart
Relative to a fixed origin OO,
  • the point AA has position vector −i+4j+3k-\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}
  • the point BB has position vector 5i+6j−k5\mathbf{i} + 6\mathbf{j} - \mathbf{k}
The line l1l_1 passes through AA and BB.
(a) Find a vector equation for l1l_1.

(2 marks)

The line l2l_2 has equation

r=(pi+12j+2k)+μ(2i+qj−k)\mathbf{r} = (p\mathbf{i} + 12\mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + q\mathbf{j} - \mathbf{k})

where μ\mu is a scalar parameter and pp and qq are constants.

Given that l2l_2 passes through BB,
(b) find the value of pp and the value of qq.

(3 marks)

The acute angle between l1l_1 and l2l_2 is θ\theta.
(c) Show that cos⁡θ=147\cos\theta = \dfrac{\sqrt{14}}{7}

(3 marks)

The point CC lies on l2l_2 such that ACAC is perpendicular to l2l_2.
(d) Using your answer to part (c), or otherwise, find the exact length of ACAC, giving your answer as a simplified surd.

(3 marks)
●●●●●Level 511 marksStart
Relative to a fixed origin OO, the line ll has equation

r=(−i+3j+4k)+λ(2i+j−2k)\mathbf{r} = (-\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})

where λ\lambda is a scalar parameter.

The point AA and the point BB lie on line ll.

Given that
  • AA has coordinates (1,a,2)(1, a, 2)
  • BB has coordinates (b,6,−2)(b, 6, -2)
(a) find the value of the constant aa and the value of the constant bb.

(2 marks)
(b) Hence show that AB→=4i+2j−4k\overrightarrow{AB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}

(2 marks)

The point CC has coordinates (3,9,5)(3, 9, 5).
(c) Find the size of angle BACBAC, giving your answer in degrees to one decimal place.

(4 marks)

The point DD lies on ll such that the area of triangle ACDACD is three times the area of triangle ABCABC.
(d) Find the coordinates of the two possible positions of DD.

(4 marks)
●●●●●Level 512 marksStart
The points AA, BB and CC have coordinates (1,0,2)(1, 0, 2), (5,8,−2)(5, 8, -2) and (7,6,k)(7, 6, k) respectively, where kk is a constant.
(a) Given that angle ACBACB is a right angle, find the possible values of kk. (3 marks)
(b) Given also that k>0k > 0, find the exact shortest distance from CC to the line ABAB. (3 marks)
(c) The point DD is the point on ABAB that is closest to CC. Find the ratio AD:DBAD : DB in its simplest form. (2 marks)
●●●●●Level 58 marksStart
Relative to a fixed origin OO, the points AA and BB have position vectors 3j−3k3\mathbf{j} - 3\mathbf{k} and 6i+3k6\mathbf{i} + 3\mathbf{k} respectively.

The point PP lies on the line segment ABAB and is such that OPOP is perpendicular to ABAB.
(a) Find the position vector of PP. (5 marks)
(b) Hence find the exact area of triangle OABOAB. (2 marks)
●●●●●Level 57 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=−i+j+4k\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}, b=5i+4j−2k\qquad \mathbf{b} = 5\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}, c=5i+7j+4k\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}.
(a) Show that angle BAC=π4BAC = \dfrac{\pi}{4}. (4 marks)

The point PP lies on the line segment ABAB and is such that angle APC=π2APC = \dfrac{\pi}{2}.
(b) Find the position vector of PP. (3 marks)
(c) Find the area of triangle ABCABC. (2 marks)
●●●●●Level 59 marksStart
Figure 2 shows the triangle OABOAB, where OA→=a\overrightarrow{OA} = \mathbf{a} and OB→=b\overrightarrow{OB} = \mathbf{b}.

The point MM is the midpoint of OBOB. The point NN lies on ABAB with AN:NB=1:3AN : NB = 1 : 3. The lines ONON and AMAM intersect at the point XX.OABMNXabFigure 2 (not accurately drawn)

Given that OX→=λ ON→\overrightarrow{OX} = \lambda\,\overrightarrow{ON} and AX→=μ AM→\overrightarrow{AX} = \mu\,\overrightarrow{AM}, where λ\lambda and μ\mu are scalars, find the value of λ\lambda and the value of μ\mu. (7 marks)
●●●●●Level 57 marksStart
Relative to a fixed origin OO, the points AA, BB and CC have position vectors

a=i+2k,b=4i+3j+5kandc=7i+6j−k.\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.
(a) Show that angle ABCABC is a right angle. (3 marks)
(b) The point DD lies on ACAC and is such that BDBD is perpendicular to ACAC. Determine the position vector of DD. (5 marks)
(c) Find the exact length of BDBD. (1 mark)
●●●●●Level 59 marksStart
Points AA and BB have position vectors 2i−j+k2\mathbf{i} - \mathbf{j} + \mathbf{k} and 4i+3j−k4\mathbf{i} + 3\mathbf{j} - \mathbf{k}. The point CC with position vector i+λj+μk\mathbf{i} + \lambda\mathbf{j} + \mu\mathbf{k} lies on the line through AA and BB. Find λ\lambda and μ\mu.
●●●●●Level 54 marksStart