MCV4U1 VECTORS CUMULATIVE REVIEW

1. Draw a regular hexagon whose vertices are labelled A, B, C, D, E, and F, in a clockwise direction. Write a single vector that is equivalent to each of the following:

a) A⃗E+ E⃗B b) A⃗C−B⃗C c) C⃗E+D⃗B+ A⃗D d) D⃗B−E⃗A−D⃗E

2. In a soccer game, two opposing players kick the ball at the same time: one with a force of 200 N straight along the sidelines, and the other with a force of 225 N directly across the field. Calculate the resultant force. Include both a vector and a position diagram.

3. A ship's course is set at a heading of N 53 °W at 18 knots. A 10-knot current flows at a

bearing of N 22°W . What is the ground velocity of the ship? Include both a vector and a position diagram.

4. A 150-N crate is resting on a ramp that is inclined at an angle of 10 ° to the horizontal. Resolve the weight of the crate into forces parallel to and perpendicular to the surface of the ramp. Include both a vector and a position diagram.

5. An airplane is flying at an airspeed of 400 km/h on a heading of S50 °W . A 45 km/h wind

is blowing from a bearing of N 30 ° E . Determine the ground velocity of the airplane. Include both a vector and a position diagram.

6. Devon is holding his father's wheelchair on a ramp inclined at an angle of 20 ° to the horizontal with a force of magnitude 2000 N parallel to the surface of the ramp. Determine the weight of Devon's father and his chair and the component of the weight that is perpendicular to the surface of the ramp. Include both a vector and a position diagram.

7. A 100-N box is held by two cables fastened to the ceiling at angles of 80 ° and 70 ° to the horizontal. a) Draw a position and vector diagram.b) Determine the tension is each cable.

8. Show that the cross product of two unit vectors is not generally a unit vector.

9. The points A (2,4 ) , B (0,0 ) and C (−2,1 ) define a triangle in a plane. Find the cosine of

∠ ABC .

10. Consider the two lines with equations

x+81

= y+43

= z−21 and

( x , y , z )=(3,3,3 )+t (4 ,−1 ,−1 ) , t∈R .

a) Show that the lines are perpendicular.b) Find the point of intersection of the lines.

11. Determine whether the origin lies on the plane that passes through the three points

P (1 ,−1,3 ) , Q (−1 ,−2,5 ) , and R (−5 ,−5,1 ) .

12. Determine the Cartesian equation of the plane that passes through the point P (6 ,−1,1 ) , has a

z-intercept of – 4, and is parallel to the line

x+23

= y+13

=−z.

13. Determine a point, A, on the line with equation r⃗=(−3,4,3 )+t (−1,1,0 ) , t∈ R , and a point,

B, on the line r⃗=(3,6 ,−3 )+s (1,2 ,−2 ) , s∈R so that A⃗B is parallel to m⃗=(2 ,−1,3 ) .

14. The equation ( x−1 )2+ ( y−2 )2+ (z−3 )2=9 defines a sphere in three-space. Find the

Cartesian equation of a the plane that is tangent to the sphere at the point (2,4,5 ) , which is a point at one end of a diameter of the sphere.

15. Determine the intersection of the line x=−1+t ¿ } y=3+2t ¿ }¿¿ t ∈R ¿

with each of the following planes:

a) x− y−z+2=0 b) −4 x+ y−2 z−7=0 c) x+4 y−3 z+7=0

16. Find the point on the xy-plane that lies on the line of intersection of the planes with

equations 4 x−2 y−z=7 and x+2 y+3 z=3 .

17. A plane passes through the points (2,0,2 ) , (2,1,1 ) , and (2,2,4 ) . A line passes through the

points (3,2,1 ) and (1,3,4 ) . Find the point of intersection of the plane and the line.

18.a) Determine the parametric equations of the line of intersection of the two planes

3 x− y+4 z+6=0 and x+2 y−z−5=0 .

b) At what points does the line of intersection intersect the three coordinate planes?c) Determine the distance between the xy-intercept and the xz-intercept.

19. Solve the system:

x+2 y−3 z=12 x+5 y+4 z=13 x+6 y−z=3

ANSWERS:

1.a) A⃗B b) A⃗B c) C⃗D d) E⃗D

2. 301.1 N at 48 . 4 ° rel. to the sideline

3. 27.1 knots [N 42 °W ]4. parallel to: 26.0 N, perpendicular to: 147.7 N

5. 442.4 km/h [S48 °W ]6. 5848 N, 5495 N7.b) 68.4 N, 34.7 N 9. 0

10b) (−5,5,5 )11. no

12. 2 x−3 y−3 z−12=0

13. A( 7

11,

411, 3)

and B( 3

11,

611,

2711 )

14.x+2 y+2 z−20=0

15.a) no intersection b) r⃗=(−1,3,0 )+t (1,2 ,−1 ) , t∈ R c) (−5

2,0 ,

32 )

16. (2 , 12 ,0)

17. (2 , 52 , 52 )

18a) x=−1−t ¿} y=3+t ¿ }¿¿ t∈ R ¿

b) xy-plane at (−1,3,0 ) , xz-plane at (2,0 ,−3 ) c)3√3units

19. ( x , y , z )=(3 ,−1,0 )