Vectors. 1. Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB).

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Jodie Byrd
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1 Vectors. Consider the points A ( ) B ( ) and D ( k ) with (AD) perpendicular to (AB). (a) Find (i) AB ; (ii) AD giving your answer in terms of k. (b) Show that k = The point C is such that BC = AD. (c) Find the position vector of C (d) Find cos ABˆ C. Let v = i + j + 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 ( ). (a) (i) Show that AB =. (ii) Find BÂO (b) The line L has equation x y s. z Write down the coordinates of two points on L. (c) The line L passes through A and is parallel to OB. (i) Find a vector equation for L giving your answer in the form r = a + tb. (ii) Point C (k k )is on L. Find the coordinates of C.

2 (d) The line L has equation and passes through the point C. Find the value of p at C.. The line L is represented by r = and the line L by r = The lines L and L intersect at point T. Find the coordinates of T.. The diagram below shows a cuboid (rectangular solid) OJKLMNPQ. The vertex O is ( ) J is ( ) K is ( ) M is ( ) and Q is ( ). (a) (i) Show that =. (ii) Find. (b) An equation for the line (MK) is r = [ ] + s [ ]. (i) Write down an equation for the line (JQ) in the form r = a + tb. (ii) Find the acute angle between (JQ) and (MK). (c) The lines (JQ) and (MK) intersect at D. Find the position vector of D. 8 p z y x s. 8 t JQ MK

3 Solution : (a) (i) evidence of combining vectors eg = (or = + in part (ii)) = (ii) = (b) evidence of using perpendicularity scalar product = (k ) + = k + 8 = (accept any correct equation clearly leading to k = ) k = (c) = = evidence of correct approach eg = (d) METHOD choosing appropriate vectors finding the scalar product eg () + () + () () + ()() + ()() AB OB OA AD AO OD AB AD k.. k e g AD BC BC OB OC z y x OC BC BA

4 cos ABˆ C = METHOD BC parallel to AD (may show this on a diagram with points labelled) BC AB (may show this on a diagram with points labelled) ABˆ C = 9 cos ABˆ C =. Solution : pw = pi + pj pk (seen anywhere) (A) attempt to find v + pw eg i + j + k + p(i + j k) collecting terms ( + p)i + ( + p)j + ( p) k attempt to find the dot product eg ( + p) + ( + p) ( p) setting their dot product equal to eg ( + p) + ( + p) ( p) = simplifying eg + p p + 9p = p + 8 = P =. 8. Solution : (a) (i)evidence of approach eg AO + OB = AB B A AB = (ii) for choosing correct vectors ( AO with AB or OA with BA ) Note: Using AO with BA will lead to.99. If they then say B Â O =.99 this is a correct solution. calculating AO AB AO AB eg d d = ()() + ()() + ()() (= 9)

5 evidence of using the formula to find the angle eg cos = =.99 radians (accept.8) (b) two correct answers eg ( ) ( ) ( ) ( ) (c) (i) r = (ii) C on L so evidence of equating components eg t = k + t = k = + t one correct value t = k = (seen anywhere) coordinates of C are ( ) (d) for setting up one (or more) correct equation using eg + p = 8 p = p = p =. Solution : evidence of equating vectors (M) eg L = L for any two correct equations eg + s = t + s = + t + s = 8 t attempting to solve the equations finding one correct parameter (s = t = ) d d BÂO t t k k 8 p

6 the coordinates of T are ( ). Solution : (a) (i)evidence of approach eg JQ = JQJOOQ JQ (ii) MK (b) (i) r = + t or r = + t Note: Award A if r = is missing. (ii) Evidence of choosing correct vectors Evidence of calculating magnitudes eg 8 8 = 9 + (= ) (accept ) For evidence of substitution into the correct formula eg cos = accept 8 8 (c) METHOD =.9 (radians) 8. Geometric approach

7 Valid reasoning eg diagonals bisect each other Calculation of mid point eg (accept (.)) METHOD Correct approach eg + t = + s Two correct equations eg t = st = s t = s Attempt to solve One correct parameter s =. t =. (accept (. )) METHOD Correct approach eg + t = + s Two correct equations eg t = s + t = s + t = s Attempt to solve One correct parameter s =. t =. MK OM OD. OD. OD

8 OD. (accept (. ))

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)