TEST1/MAC2313
Page 3 of 5
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9. (5 pts.) Suppose
v
= <-3,-4, 5> and
w
= <-1,1,1>.
If
α
,
β
, and
γ
are the direction angles of
v
, then
cos(
α
) =
,
cos(
β
) =
,and
cos(
γ
) =
.
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10. (5 pts.) Suppose
v
= <-3,-2, 1> and
w
= <-1,1,1>.
What is the exact
value of the angle
θ
between
v
and
w
??
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11. (5 pts.) Write a point-normal equation for the plane perpendicular to
v
= <-3,-2,1> and containing the point
(-1,2,-3).
-3(x - (-1)) - 2(y - 2) + (z - (-3)) =
0
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12. (5 pts.) Which point on the line defined the vector equation
<x,y,z> = <1,1,1> + t<2,-1,-1> is nearest the point, (0,-1,0)?
Build the vector with initial point (0,-1,0) to an arbitrary point on
the line with parameter
t
,
The norm of this vector is the distance from the point (0,-1,0) to the
point on the line with parameter value
t
.
The norm will be smallest when
the vector is perpendicular to the vector <2,-1,-1>.
Consequently,
provides us with the value of
t
needed for the closest point.