Math 210 Take Home Assignment #1
Due: Thursday, September 17
Unless otherwise indicated, each question is worth 4 points. Please show your work in the space provided if at
all possible.
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1) Find a vector with a magnitude of 4 and in the same direction as v  2,1,3  .
2) Write the equation of the sphere in standard form. Identify the center and the radius for the sphere:
x 2  y 2  z 2 9 x  2 y  10 z  19  0
3) Given initial point P(4, 3, 0) and terminal point Q(1, –3, 3), find the point that lies two-thirds of the way
from P to Q.
4) (2 points) Plot the following points in space:
a) (2, 1, 5)
b) (3, 1, −2)
c) (−1, 2, −4)
d) (−3, −4, 0)
5) An airplane has an airspeed of 400 mph. Suppose that the wind velocity is given by the vector
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w   20,30 . In what direction should the airplane head in order to fly due west (i.e. in the direction of
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the unit vector  i ).
6) Find the component form and the magnitude of a vector having initial point P(−2, 3, 1) and terminal
point (0, −4, 4). Sketch the vector (with initial point at the origin). Find a unit vector in the direction of
the given vector.
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7) (8 points) Given: u  2,1,2  and v  0,3,4  , find the following:
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a) u  v
d)
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4u  5v
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4u  5v
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g) angle between u and v
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b) u  v
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c) 4u  5v
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e) u  v
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f) proj v u
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h) unit vector that is orthogonal to both u and v
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8) a) Find the area of a parallelogram with vectors a  3i  2 j  k and b  i  2 j  3k as adjacent sides.
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b) Find the volume of the parallelepiped with the given vectors as adjacent edges: a  i  2 j  3k ,
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b  4i  5 j  6k , c  7i  8 j
9) Find the sets of parametric equations and symmetric equations for the line that passes through the points
P(1, 2, −1) and Q(5, −3, 4).
10) Find the equation of the plane containing the point P(1, 2, 3) with normal vector <4, 5, 6> and sketch the
plane (use the traces we discussed in class).