Que.1 [A] Do the following multiple choice questions. 1. A line in two-space has scalar equation 3+ 5y + 9 = 0. Which vector is parallel to the line? a. [3,5] b. [3.-5] c. [5,3] d. [-5,3] 2. Which plane does not contain the point P(10, -3, 5)? a. 3x+6y-2z-2=0 b. X+y-z-12=0 c. 2x-2y-3z-11=0 d. 4x+5y+z-30=0 3. How many solutions does this system have? 10x -7y = 10 4x + 5y = 12 a. 0 b. 1 c. 12 d. Infinitely many

3.Convert each true bearing to its equivalent quadrant bearing. g. 130 h. 080 4. Write a force of 200 N at 20to the horizontal in Cartesian form. ??????? 5. Find the coordinates and magnitude of vector?? ,for A(1 ,3) and B(7,2). 6. Calculate the dot product for the followings. a. ??? = [2 ,4] ??? = [3 ,-1] b.???=2i+3j ??? =9i-7j 7. calculate the angle between the vectors ??? = [7 ,8] and ??? = [4 ,3]

3.Convert each true bearing to its equivalent quadrant bearing. g. 130 h. 080 4. Write a force of 200 N at 20to the horizontal in Cartesian form. ??????? 5. Find the coordinates and magnitude of vector?? ,for A(1 ,3) and B(7,2). 6. Calculate the dot product for the followings. a. ??? = [2 ,4] ??? = [3 ,-1] b.???=2i+3j ??? =9i-7j 7. calculate the angle between the vectors ??? = [7 ,8] and ??? = [4 ,3] 8. determine the projection of ??? and ??? if I??? I = 56 , I??? I = 100 ,angle between ??? and ??? is 125.

Que.2 Do the following sums. (T/I) 1. Determine the value of k so that ??? = [2 ,5] and ??? = [k , 4] are perpendicular.

2. Write a scalar equation of a plane, given its parametric equations. ?1 : x = 3 -2s +2t , Z = 5 - s -2t Y = 1 + 3s + t 3. Considertheplanes2x+6y-2z-5=0and5x+15y+kz-7=0 a. For what value of k are the planes parallel ? b. For what value of k are the planes perpendicular?

4. A tow truck is pulling a car from a ditch. The tension in the cable is 15 000 N at an angle of 40 to the horizontal. a) Draw a diagram showing the resolution of the force into its rectangular components b) Determine the magnitudes of the horizontal and vertical components of the force. 5. find b and c such that [-2 ,b,7] and [c ,6 ,21] are collinear. 5. if ??? = [6 , 1 ,8] is orthogonal to ??? = [k ,-4 ,5] determine the value(s) of k. 6. Show that the line [x ,y ,z] = [10, 5 ,16] + t[3,1 ,5] is contained in the plane x + 2y - z -4 =0

Que.3 Do the following sums.(C) 1. Describe the differences between airspeed , windspeed , and groundspeed when solving vector problems associated with airplane flight. 2. Which form of the equation of the plane?scalar, vector, or parametric? would you use to find the x-, y-, and z-intercepts? Explain your reasons.

3. When algebraically determining the intersection of a plane and a line, the result is 0 = 0. What does this tell you about the plane and line? Justify your answer.

4. Explain what the term consistent means in the context of a system of three planes. 5. Find the point of intersection of lines l1 and l2. l1 : [x ,y ,z] = [7 ,2 ,-6] + s[2 ,1,-3] 6. Explain how the vectors ??? , 5??? , and -5??? are related. 7. Draw a diagram resolving a 500-N force at 15 counterclockwise from the horizontal into its rectangular components. 8. What is the difference between [2,3] and (2,3) ?

9. Write some application of the dot product measure? 10. Describe the method you would use to plot the point (-3, 5,-7). Que.4 Do the following sums.(A) 1. An airplane is flying at an airspeed of 500 km/h, on a heading of 040. A 150-km/h wind is blowing from a bearing of 120. Determine the ground velocity of the airplane and the direction of flight. 2. Determine the following system is consistent or inconsistent. if possible ,solve the system. 3x+y-2z=18 ,6x-4y+10z=-10 ,3x-5y+10z=10

3. It is important for aerospace engineers to know the vertical and horizontal components of a rocket's velocity. If a rocket is propelled at an initial velocity of 120 m/s at 80 from the horizontal, what are the vertical and horizontal components of the velocity? 4. Find the volume of parallelepiped, defined by the vectors ??? = [1, 4, 3], ??? = [2, 5, 6], and ??? =[1,2,7]. 5. A wrench is used to tighten a bolt. A force of 60 N is applied in a clockwise direction at 80 to the handle, 20 cm from the centre of the bolt. a) Calculate the magnitude of the torque. b) In what direction does the torque vector point? 6. Use the normal vectors of the planes to describe the following system. 2x+5y+3z=0 , x-3y+6z=19 ,3x+2y+9z=-7

Que.1[B] do the following sums : 1. Use the following set of vectors to draw a diagram of a + b- c. 2.In hexagon ABCDEF, opposite sides are parallel and equal, FC =2AB . Let AB = u and FA = v A B D Express the following vector in terms of u and v e. CF f. FD