MATH 127
Vector Analysis
Triple Product
1. →A.(→B×→C)
2. →A×(→B×→C)
3. (→A.→B).→C
• Projection:
i) Scaler Projection: →b→a|→a|
ii) Vector Projection: →a.→b|→a|2→a
The scaler projection of
→b
and
→a
is the length of the AB (shown in the figure). The vector projection of
→b
onto
→a
is the vector with this length at the point A point in the same direction as
→a
.
This quantity is also called the component of
→b
in the
→a
direction. And, the vector projection is merely the unit vector
→a|→a
times the scalar projection of
→b
onto
→a
.
∴ The scalar projection of →b onto →a is proj →b→a=→b.→a→a
and the vector projection of →b onto →a is proj →b→a=→a.→b|→a|2→a .
♦ Direction Cosines:
cosα=x|→r|=x√x2+y2+z2
cosβ=y|→r|=y√x2+y2+z2
cosγ=z|→r|=z√x2+y2+z2
Theorem: cos2α+cos2β+cos2γ=1
Example: Find direction cosines of a vector represented by
→PQ
where
P(2,−3,5) and Q(1,0,−1)
are two point.
Solution: →PQ=(1−2)ˆi+(0+3)ˆj+(−1−5)ˆk=−ˆi+3ˆj−5ˆkNow,α=cos−1(−1√12+32+62)=cos−1(−1√46) =98.47∘β=cos−1(3√12+32+62)=cos−13√46 =63.74∘θ=cos−1(−6√12+32+62)=cos−16√46 =152.2∘
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