Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises: 43

$\frac{-7}{\sqrt {19}}$, $\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$

Work Step by Step

Given: $a=3i-3j+k$ , $b=2i+4j-k$ Change the form of $a$ and $b$. $a=\lt3,-3,1\gt$ , $b=\lt2,4,-1\gt$ Scalar Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|}=\frac{(3 \times 2)+( -3 \times 4)+(1 \times -1)}{\sqrt {{(3)^{2}+(-3)^{2}}+(1)^{2}}}$ $=\frac{6-12-1}{\sqrt {19}}$ $=\frac{-7}{\sqrt {19}}$ Vector Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|^{2}}\times a=\frac{-7}{19}\lt3,-3,1\gt$ $=\lt\frac{-21}{19}, \frac{21}{19},\frac{-7}{19}\gt$ Change vector projection back into $i+j+k$ form. $=\frac{-21}{19}i+ \frac{21}{19}j+\frac{-7}{19}k$ $=\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$ Hence, Scalar Projection $b$ onto $a$ = $\frac{-7}{\sqrt {19}}$, Vector Projection $b$ onto $a$=$\frac{-21}{19}i+ \frac{21}{19}j-\frac{7}{19}k$

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