Vector names are in bold (u), scalars (constants) are italics (a).
Suggestion: to indicate vectors (since you can't make the text bold) --
write them like this
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| LINEAR ALGEBRA PRACTICE EXERCISES VECTORS |
Vectors are displayed vertically to distinguish them from points.
Vector names are in bold (u), scalars (constants) are italics (a).
Suggestion: to indicate vectors (since you can't make the text bold) --
write them like this .
QUESTIONS
1) Find cos h if h is the angle between u and v if:
a)  |
b)  . |
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2) Prove that u is orthogonal to v if 
.
3) Find yuyif:
a)  |
b)  |
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4) Find a vector v that is parallel to vector 
and yvy = 1.
.
5)
, show that any vector orthogonal to x is a scalar multiple of 
.
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6) 
a) Prove that A is a set of orthogonal vectors.
b) Prove that if x is any vector orthogonal to both vectors in A,
then x is a scalar multiple of 
.
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7) 
| a) Find proj w u. | b) Find the component of u orthogonal to w. |
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SOLUTIONS
1)
a)  |
b)  . |
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2) u is orthogonal to v because u $ v = 0. u $ v = -12 + 8 + 4 = 0
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3)
a)  |
b)  |
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4) Since v is parallel to 
it is a multiple of u or ku.
and since yvy = 1, k = 1 / yvy. This makes 
.
5) If v is orthogonal to x, then v $ x = 0
If we let x2 = t, we get x1 = 2t, so v = 
.
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6) a) 
Since dot product = 0, vectors are orthogonal.
b) A vector orthogonal to both vectors in A is the cross-product of the vectors.
The cross-product is 
.
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7) 
a) |
b) The component of u orthogonal to w is found with u - proj w u. the vector is |
.
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