assign1_soln - Math 136 Assignment 1 Solutions 1 Compute...

Math 136 Assignment 1 Solutions
1. Compute each of the following.
a) (1 , 3 , 4) + ( - 1 , 1 , 2)
Solution: (1 , 3 , 4) + ( - 1 , 1 , 2) = (0 , 4 , 6).
b) 3( - 1 , 1 , - 2) - 2(2 , 0 , 3).
Solution: 3( - 1 , 1 , - 2) - 2(2 , 0 , 3) = ( - 3 , 3 , - 6) + ( - 4 , 0 , - 6) = ( - 7 , 3 , - 12).
2. Determine the distance between P (2 , 1 , 1) and Q (1 , - 1 , 1).
Solution: The distance is (1 , - 1 , 1) - (2 , 1 , 1) = ( - 1 , - 2 , 0) = ( - 1)
2
+ ( - 2)
2
+ 0
2
=
√
5.
3. Determine which of the following pairs of vectors is orthogonal.
a) (2 , 1 , 1), ( - 2 , 1 , 3).
Solution: We have (2 , 1 , 1) · ( - 2 , 1 , 3) = - 4 + 1 + 3 = 0 so they are orthogonal.
b) ( - 1 , 3 , 6), (3 , - 1 , 0).
Solution: We have ( - 1 , 3 , 6) · (3 , - 1 , 0) = - 3 - 3 + 0 = - 6 so they are not orthogonal.
4. Find an equation for the plane through P (3 , - 2 , 1) and parallel to x
1
- x
2
+ 2 x
3
= 4.
Solution: Since the plane is parallel to x
1
- x
2
+ 2 x
3
= 4 it must have normal vector n =
(1 , - 1 , 2). Thus the equation of the plane is
x
1
- x
2
+ 2 x
3
= 1(3) - 1( - 2) + 2(1) = 7 .
5. For each of the following sets:
i) Determine if the set is linearly dependent or linearly independent. Justify.
ii) Describe geometrically the span of the set and give a simplified vector equation
which describes it.