Parallel or perpendicular vectors condition

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin \(O\).] A particle \(P\) is moving in a straight line with constant velocity. At 9 am, the position vector of \(P\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) and at 9.20 am , the position vector of \(P\) is \(6 \mathbf { i } \mathrm {~km}\). At time \(t\) hours after 9 am , the position vector of \(P\) is \(\mathbf { r } _ { P } \mathrm {~km}\).

1. Find, in \(\mathrm { kmh } ^ { - 1 }\), the speed of \(P\).
2. Show that \(\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }\).
3. Find the value of \(t\) when \(\mathbf { r } _ { P }\) is parallel to \(16 \mathbf { i } + 5 \mathbf { j }\). The position vector of another particle \(Q\), at time \(t\) hours after 9 am , is \(\mathbf { r } _ { Q } \mathrm {~km}\), where \(\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }\)
4. Show that \(P\) and \(Q\) will collide and find the position vector of the point of collision.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\) ]

A particle \(P\) is moving with velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\) hours, the position vector of \(P\) is \(( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).

1. Find an expression for \(\mathbf { p }\) in terms of \(t\). The point \(A\) has position vector ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) km.
2. Find the position vector of \(P\) when \(P\) is due west of \(A\). Another particle \(Q\) is moving with velocity \([ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }\) where \(b\) is a constant. Given that the particles are moving along parallel lines,
3. find the value of \(b\).

3 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } )\) N. Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).

5 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$

1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.

5 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.

1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
2. Find \(k\).

4

1. Simplify \(2 \binom { 6 } { - 3 } - 3 \binom { - 1 } { 2 }\).
2. The vector \(\mathbf { a }\) is defined by \(\mathbf { a } = r \binom { 6 } { - 3 } + s \binom { - 1 } { 2 }\), where \(r\) and \(s\) are constants. Determine two pairs of values of \(r\) and \(s\) such that \(\mathbf { a }\) is parallel to the \(y\)-axis and \(| \mathbf { a } | = 3\).

3. In a simple model for the motion of a car, its velocity, \(\mathbf { v }\), at time \(t\) seconds, is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 2 t + 8 \right) \mathbf { i } + ( 5 t + 6 ) \mathbf { j } \mathrm { ms } ^ { - 1 }$$

1. Calculate the speed of the car when \(t = 0\).
2. Find the values of \(t\) for which the velocity of the car is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).
3. Why would this model not be appropriate for large values of \(t\) ?

4. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and \(O\) is a fixed origin. A pedestrian moves with constant velocity \(\left[ \left( 2 q ^ { 2 } - 3 \right) \mathbf { i } + ( q + 2 ) \mathbf { j } \right] \mathrm { ms } ^ { - 1 }\). Given that the velocity of the pedestrian is parallel to the vector \(( \mathbf { i } - \mathbf { j } )\),

1. Show that one possible value of \(q\) is \({ } ^ { - } 1\) and find the other possible value of \(q\). Given that \(q = { } ^ { - } 1\), and that the pedestrian started walking at the point with position vector \(( 6 \mathbf { i } - \mathbf { j } ) \mathrm { m }\),
2. find the length of time for which the pedestrian is less than 5 m from \(O\).

3 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.

1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
2. Find \(k\).

7 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } ) \mathrm { N }\). Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).

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

16 Two particles, \(P\) and \(Q\), move in the same horizontal plane. Particle \(P\) is initially at rest at the point with position vector \(( - 4 \mathbf { i } + 5 \mathbf { j } )\) metres and moves with constant acceleration \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\) Particle \(Q\) moves in a straight line, passing through the points with position vectors \(( \mathbf { i } - \mathbf { j } )\) metres and \(( 10 \mathbf { i } + c \mathbf { j } )\) metres. \(P\) and \(Q\) are moving along parallel paths.
16

1. Show that \(c = - 13\) 16
2. (i) Find an expression for the position vector of \(P\) at time \(t\) seconds.
   16 (b) (ii) Hence, prove that the paths of \(P\) and \(Q\) are not collinear.

Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}$$

1. Find the value of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [2]
2. Find the values of \(p\) for which the magnitude of \(\overrightarrow{AB}\) is 7. [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]

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]

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]