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Let vector A=a1i +a2j + a3k and B=b1i + b2j + b3k ne on the same plane. Fine the unit vector perpendicular to both A and B.

Find the position vector of the point in space P that lies on the line AB such that |AP|/|PB|= m/n with m, n is a real number. Find the position vector of P if vector A=(1,2,1), B(3,-1,2), m=3 and n=2

Three points with position vectors, b and c are said to be colinear. If the parallelogram with adjacent sides a - b and a - c has zero geometry area. Use this fact to check whether or not the following triples of points are collinear

(a) (2,2,3), (6,1,5) (2,4,3)

(b) (2,3,3), (3,7,5), (0,-5,-1)

(c) (1,3,2), (4,2,1), (1,0,2)

1. A hyperbola has vertices (1,9) and (14,9) and one of its foci is (-2,9) find its standard equation.

2. Determine the foci vertices and asymptote of the hyperbola with equation.

x²/16-y²/20=1

Sketch the graph and include these points and lines along with the auxiliary rectangle.

3. Give the coordinates of the foci vertices and asymptote of the hyperbola with equation 9x2-4y2-90x-32y=-305. Sketch the graph and include these and lines along 2ith auxiliary rectangle.

7. An aeroplane heads due north at 500 km/h. It experiences a 80 km/h crosswind flowing in

the direction N60oE.

(a) Find the true velocity of the aeroplane. (7)

(b) Determine the speed of the aeroplane. (Leave your answer in terms of square root

6. Four forces act on an object such that the object is at rest. Three of the forces are given by

F1 = 2i −2j, F2 = i −4j, F4 = −3i −5j. Determine F3 and its magnitude

Find the centroid of the triangle whose vertices are (−1,0), (1,0), 𝑎𝑛𝑑 (0,3

Determine the moment vector m about the origin, o, of a force f= 10i-j+2k passing through the point with position vector r=2i-3j+5k

Convert the following rectangular coordinates into polar coordinates (r,θ) so that r < 0

and 0 ≤ θ ≤ 2π :

(4,−4√3)

. (a) Plot the following points in the same polar coordinates system

(3,π/4),(−3,π/4),(3,−π/4),(−3,−π/4)

(b) Convert into rectangular coordinates:

(4,−2π/3)

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