math 2110/ routine questions and computations from chapter 9
1. Given two vectors, how would you determine if they are orthogonal ? How would you
determine if one vector had the same direction as another?
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2. Given two vectors a and b which are not multiples of each other, how many unit vectors
are orthogonal to both of these vectors? How would you go about computing them ?
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3. How does one define the amount of work which is done when a force F of constant
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magnitude is applied to an object, moving it in the direction d for a distance | d |. Is work a
scalar or a vector?
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4. Under what circumstances will a x a  0 ? Under what circumstances will a a  0 ?
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5. The vectors a and b are drawn below.
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a. Draw the vectors a + b , a - b , 3 a , and -0.5 a .
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b. What is the direction of a x b ? What is the direction of b x a ?
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a
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b
6. If two distinct vectors are orthogonal to a plane are they necessarily scalar multiples of
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each other? If two distinct vectors a and b are orthogonal to a third vector c , are a and
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b necessarily orthogonal to each other?
7. You should be able to find:
a. the equation of a line, given two distinct points on the line.
b. the equation of a line, given a point and a direction vector.
c. the equation of a plane, given three points (not colinear) lying in the plane.
d. the equation of a plane, given a normal vector to the plane and a point in the plane.
e. the dot and cross product of two vectors.
8. Given the equation of a line you should be able to:
a. provide a vector in the direction of the line, regardless of whether it is in
parametric or symmetric form.
b. convert the equation from parametric form to symmetric form and vice versa.
11. You should be able to determine:
a. if two lines are parallel, skew, or intersecting. If they intersect, you should be able
to determine the point of intersection and the angle between the lines. If parallel
or skew, you should be able to demonstrate why that is the case.
b. if two planes are parallel or intersecting. If the planes intersect, you should be
able to determine the line of intersection and the angle between the planes.
c. if two vectors are parallel, orthogonal, or neither. In the latter case, you should
be able to find the angle between the vectors.
d. the magnitude of a vector and the unit vector going in the same
direction as the original vector.
e. the magnitude and direction angle of a vector given in component form. If you
know the magnitude and direction of a vector, you should be able to write it in
component form.
f. the scalar and vector projection of one vector onto another vector.
g. the distance from a point to a plane and the distance from a point to a line.