(a) Sketch vectors u and v for which u + v and u − v are orthogonal. (b) How can you use vectors to determine whether four points in 3-space lie in the same plane? (c) If forces F 1 = i and F 2 = j are applied at a point in 2-space, what force would you apply at that point to cancel the combined effect of F 1 and F 2 ? (d) Write an equation of the sphere with center 1 , − 2 , 2 that passes through the origin.
(a) Sketch vectors u and v for which u + v and u − v are orthogonal. (b) How can you use vectors to determine whether four points in 3-space lie in the same plane? (c) If forces F 1 = i and F 2 = j are applied at a point in 2-space, what force would you apply at that point to cancel the combined effect of F 1 and F 2 ? (d) Write an equation of the sphere with center 1 , − 2 , 2 that passes through the origin.
(a) Sketch vectors u and v for which
u
+
v and u
−
v
are orthogonal.
(b) How can you use vectors to determine whether four points in 3-space lie in the same plane?
(c) If forces
F
1
=
i and F
2
=
j
are applied at a point in 2-space, what force would you apply at that point to cancel the combined effect of
F
1
and F
2
?
(d) Write an equation of the sphere with center
1
,
−
2
,
2
that passes through the origin.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Q1(A): [9 Marks] Two vectors A and B are located at P1(2,2,2) and P2(2,-2,-2). Write vector
notation for each starting from the origin. What will be the distance between both
vectors? Determine the dot product and the angle between them.
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How do you find a vector perpendicular to a plane?
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