Question

Find the scalar and vector projections of b onto a. a=(4,7,-4),b=(3,-1,1)

Vectors
ANSWERED
asked 2021-05-17
Find the scalar and vector projections of b onto a.
\(a=(4,7,-4),b=(3,-1,1)\)

Expert Answers (1)

2021-05-18
Find the dot product
\(a\cdot b=(4,7,-4)\cdot(3,-1,1)\)
\(=4\cdot3+7\cdot(-1)+(-4)\cdot1\)
\(=12-7-4\)
\(=1\)
Find the magnitude of a
\(|a|=\sqrt{4^2+7^2+(-4)^2}\)
\(=\sqrt{16+49+16}\)
\(=\sqrt{81}\)
\(=9\)
Scalar projection of b on to a is given by
\(\text{comp}_ab=\frac{a\cdot b}{|a|}\)
Substitute the values of the \(a\cdot b\) and |a|, to get
\(\text{comp}_ab=\frac{1}{9}\)
Vector projection of b onto a is given by
\(\text{proj}_ab=[\text{comp}_ab]\frac{a}{|a|}\)
Substitute the values of the \(\text{comp}_ab\) and |a|, to get
\(\text{proj}_ab=[\frac{1}{9}]\frac{(4,7,-4)}{9}=(\frac{4}{81},\frac{7}{81},-\frac{4}{81})\)
Result: \(\text{comp}_ab=\frac{1}{9}\quad\text{proj}_ab=(\frac{4}{81},\frac{7}{81},-\frac{4}{81})\)
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