Exam Style C4 Vectors Questions - Solutions

Transcription

1 Exam Style C4 Vectors Questions - Solutions 1) a) We use the equation (As ) (Simplifying the direction vector) or written as b) To find the coordinates where OC lies to r, we use the direction component of the vector equation r Use (Expanding and solving for λ) Therefore, the coordinates of C are ELITE Tuition (2009) 1

2 c) It is helpful to draw a diagram to visualise how best to answer the question: B A r O D We can see that the vector is the same as the vector Thus d) Again, drawing a diagram can help us visualise the best way to find the area C B A r O D One can see that the area of the parallelogram is the same as the square of length and height Thus the Area = Thus Area = ELITE Tuition (2009) 2

3 2) a) By definition, the points A and B must lie on the line r Thus point B lies on r By comparing the coefficients of i, j & k we get the equations: [1] [2] [3] Using equation [3] we can solve for λ Using this in equations [1] and [2] we find that: b) If point P is such that, then we use We dot the vector equation for r with the direction component of r (Substituting back into vector equation for r) Thus ELITE Tuition (2009) 3

4 c) Drawing a diagram: A P B r O The Area of a triangle is Thus: Thus: ELITE Tuition (2009) 4

5 3) a) At point B the vector equations for the lines are equal Thus: (This leads us to the equations) [1] [2] [3] Generally, one solves equations [1] & [2] simultaneously and substitute into equation [3] to find the values of λ and µ Fortunately, this question has an easy route Taking equation [3]: Substituting this into the line equation for l 1 b) To find the angle, we use the equation: It is important to note that the vectors that are dotted together must be the direction components of the lines only Let a equal the direction component of l 1, Let b equal the direction component of l 2, Continued overleaf ELITE Tuition (2009) 5

6 Thus c) (Substituting the vectors for a and b) (Substituting the vectors for c and b) Since both and equal, ELITE Tuition (2009) 6

7 d) This question is remarkably easy to answer and the technique used to answer this question arises often But as always, it is a good idea to draw a diagram l 2 C D B A l 1 O From the diagram above, it is easy to see that the position vector of D or one prefers is simply: if Thus, ELITE Tuition (2009) 7

8 4) a) We are given the vectors: & Given, b) If OACB is a rectangle, then it means that: (As they meet at a right angle) & (As they also meet at a right angle) We can find & : Thus trying : (Thus they are at right angles) (Thus they are at a right angle) Thus the exact area would be : ELITE Tuition (2009) 8

9 c) d) We need to find the vectors & Thus the angle would be: ELITE Tuition (2009) 9

10 5) a) The line is given by the equation Thus: b) One can write line l 2 as: If point A lies on l 2 then: Be equating coefficients we arrive at three equations: 1) 2) 3) Since for all three coefficients, point A exists on the line l 2 c) Using the formula Using only the direction components of the two lines: Continued overleaf ELITE Tuition (2009) 10

11 d) It is often best to draw a diagram to visualise how best to answer this question l 1 A 195 Distance C l 2 O There are many ways in which one can answer this question, but the best way to answer this question is to use trigonometry Since the shortest line linking point C to the line l 1 will meet the line l 1 at a right angle, we have a right angle triangle Thus: Distance = (Thus determining ) units Distance = Distance = 100 units ELITE Tuition (2009) 11

12 6) a) Lines l 1 and l 2 meet at point Q such that: Be equating coefficients we arrive at three equations: 1) 2) (Solving Equations 1 and 2) 3) (Substituting into Equation 2) To confirm that the two lines actually do intersect, we test these two values in Equation 3 Since the above equation is true, we have confirmed that the two lines do in fact, intersect To find the coordinates, we substitute either µ or λ: Continued overleaf ELITE Tuition (2009) 12

13 b) If two vectors are perpendicular, Taking only the direction components: Thus, the two lines are perpendicular c) It is best to draw a diagram to visualise the problem: l 1 Q P R l 2 O The area of the triangle is: Area = To find the full coordinates of P we need to determine the value of λ: Therefore: To find the full coordinates of R we need to determine the value of µ Continued overleaf ELITE Tuition (2009) 13

14 Therefore: Substituting in these values: END ELITE Tuition (2009) 14

15 Blank Page ELITE Tuition (2009) 15

16 Blank Page ELITE Tuition (2009) 16