Vectors Practice [296 marks]

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1 Vectors Practice [96 marks] The diagram shows quadrilateral ABCD with vertices A(, ), B(, 5), C(5, ) and D(4, ) a 4 Show that AC = ( ) Find BD (iii) Show that AC is perpendicular to BD The line (AC) has equation r = u + sv b Write down vector u and vector v Find a vector equation for the line (BD) c The lines (AC) and (BD) intersect at the point P(, k) Show that k = d The lines (AC) and (BD) intersect at the point P(, k) Hence find the area of triangle ACD The line passes through the points P(, 4, ) and Q(4, 5, 4) L a Find PQ Hence write down a vector equation for in the form r = a + sb L

2 p The line L is perpendicular to L, and parallel to p, where p Z 4 b Find the value of p Given that L passes through R(, 6, 4), write down a vector equation for L c The lines L and L intersect at the point A Find the x-coordinate of A A line L passes through A(,, ) and is parallel to the line r = + s 5 a Write down a vector equation for L in the form r = a + tb The line L passes through point P when t = b Find OP ; OP The following diagram shows the obtuse-angled triangle ABC such that AB = and AC = 4 6 4a Write down BA Find BC 4b Find cosabˆ C Hence, find sinabˆ C 4c 4 The point D is such that CD = 5, where p > p Given that CD = 5, show that p = Hence, show that CD is perpendicular to BC

3 4 The following diagram shows quadrilateral ABCD, with AD = BC, AB = ( ), and AC = ( ) 4 5a Find BC 5b Show that BD = ( ) 5c Show that vectors BD and AC are perpendicular Line L passes through points A(,, 4) and B(,, 5) 6a Find AB 6b Find an equation for L in the form r = a + tb Line L has equation r = 4 + s 7 Find the angle between L and 6c L 6d The lines L and L intersect at point C Find the coordinates of C A particle is moving with a constant velocity along line L Its initial position is A(6,, ) After one second the particle has moved to B( 9, 6, 5) 7a Find the velocity vector, AB Find the speed of the particle 7b Write down an equation of the line L

4 The diagram shows a parallelogram ABCD The coordinates of A, B and D are A(,, ), B(6, 4,4 ) and D(, 5, 5) a 5 Show that AB = Find AD 6 (iii) Hence show that AC = 5 Find the coordinates of point C b c Find AB AD Hence find angle A Hence, or otherwise, find the area of the parallelogram d The line L is represented by the vector equation r = + p 5 A second line L is parallel to L and passes through the point B(, 5, 5) Write down a vector equation for L in the form 9a r = a + tb 9b 5 7 A third line L is perpendicular to L and is represented by r = + q k Show that k = The lines L and intersect at the point A 9c L Find the coordinates of A 9d 6 The lines L and L intersect at point C where BC = 4 Find AB Hence, find AC

5 6 Let AB = and AC = a Find BC b Find a unit vector in the direction of AB c Show that AB is perpendicular to AC a Let u = and w Given that u is perpendicular to w, find the value of p = p b Let v = Given that, find the possible values of q v = 4 q 5 Consider the points P(,, 5) and Q(,, ) Let be the line through P and Q L a Show that PQ = [ mark] b The line L may be represented by r = + s What information does the vector give about L? Write down another vector representation for L using The point T(, 5, p) lies on L c Find the value of p d The point T also lies on L with equation y = 9 + t z q Show that q = e Let θ be the obtuse angle between L and L Calculate the size of θ

6 The vertices of the triangle PQR are defined by the position vectors OP = 4 6, OQ = and OR = 5 a Find PQ ; PR b Show that cosrpq = c Find sinrpq Hence, find the area of triangle PQR, giving your answer in the form a Two lines with equations r = + s and r = + t 5 intersect at the point P Find the coordinates of P Find the cosine of the angle between the two vectors i + 4j + 5k and 4i 5j k 5 The line L is parallel to the z-axis The point P has position vector and lies on L Write down the equation of L in the form 6a r = a + tb 6b 6 The line L has equation r = 4 + s The point A has position vector 5 9 Show that A lies on L Let B be the point of intersection of lines L and 6c L Show that OB = 4 Find AB The point C is at (,, 4) Let D be the point such that ABCD is a parallelogram 6d Find OD 7 Let v = and, for The angle between v and w is w = k 6 π k > 4 Find the value of k

7 In this question, distance is in metres Toy airplanes fly in a straight line at a constant speed Airplane passes through a point A Its position, p seconds after it has passed through A, is given by y = 4 + p z a Write down the coordinates of A Find the speed of the airplane in ms b After seven seconds the airplane passes through a point B Find the coordinates of B Find the distance the airplane has travelled during the seven seconds Airplane passes through a point C Its position q seconds after it passes through C is given by c y = 5 + q,a R z a The angle between the flight paths of Airplane and Airplane is 4 Find the two values of a 9 Let v = i + 4j + k and w = i + j k The vector v + pw is perpendicular to w Find the value of p The point O has coordinates (,, ), point A has coordinates (,, ) and point B has coordinates (, 4, ) a 4 Show that AB = 6 Find BAO [ marks] b 4 The line L has equation y = 4 + s 6 z Write down the coordinates of two points on L c The line L passes through A and is parallel to OB Find a vector equation for, giving your answer in the form r = a + tb Point C(k, k,5) is on L Find the coordinates of C L d The line L has equation y = + p and passes through the point C z Find the value of p at C Consider the points A (, 5, 4), B (,, ) and D (, k, ), with (AD) perpendicular to (AB) Find a AB ; AD giving your answer in terms of k

8 b Show that k = 7 c The point C is such that BC = AD Find the position vector of C d Find cosabc The line L is represented by r = 5 + s and the line L by r = + t 4 The lines L and L intersect at point T Find the coordinates of T In the following diagram, u = AB and v = BD BD The midpoint of AD is E and = Express each of the following vectors in terms of u and v DC a AE b EC Consider the lines L, L, L, and L 4, with respective equations L : y = + t z L : y = + p z L : y = + s z a L 4 6 : y = q 4 z Write down the line that is parallel to L 4a 4 [ mark] 4b Write down the position vector of the point of intersection of L and L [ mark]

9 4c Given that L is perpendicular to L, find the value of a The line L passes through the points A(,,4) and B(,,5) 5a Show that AB = [ mark] 5b Hence, write down a direction vector for L ; [ mark] 5c Hence, write down a vector equation for L 4 Another line L has equation r = 7 + s The lines L and L intersect at the point P 4 Find the coordinates of P 5d 5e Write down a direction vector for L [ mark] 5f Hence, find the angle between L and L International Baccalaureate Organization 6 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for North Hills Preparatory

2 and v! = 3 i! + 5 j! are given.

1. ABCD is a rectangle and O is the midpoint of [AB]. D C 2. The vectors i!, j! are unit vectors along the x-axis and y-axis respectively. The vectors u! = i! + j! 2 and v! = 3 i! + 5 j! are given. (a)