Question

3. Find the vector projection of u = 6i +3j+2j c
onto v =i-2j - 2k and the scalar component of u in the direction of v​

Answer 1

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Answer 2

Given:

u = 6i +3j+2j c

onto v =i-2j - 2k

Find:

vector projection

scalar component of u in the direction of v​

Solution:

=> Vector projection of u and v is

$\frac{\mathrm{a} \cdot \mathrm{b}}{|\mathrm{b}|}=\frac{(6 \mathrm{i}+3 \mathrm{j}+2 \mathrm{k}) \cdot(\mathrm{i}-2 \mathrm{j}-2 \mathrm{k})}{|\mathrm{i}-2 \mathrm{j}-2 \mathrm{k}|}$

$=\frac{6\time1-3\times2-2\times2}{\sqrt{(1)^{2}-(2)^{2}-(2)^{2} } } \\\\=\frac{6-6-4}{\sqrt{1-4-4} } \\\\=\frac{-4}{\sqrt{1-8} }\\\\=\frac{-4}{\sqrt{-7} }$

=> scalar component of u in the direction of v​

$proj_{v}u =( \frac{\mathrm{u} \cdot \mathrm{v}}{|\mathrm{v}|^{2} })v=\frac{-4}{\sqrt{(-7)^{2} } }(i+j+k) \\\\=\frac{-4}{7 }(i+j+k)$

Hence  vector projection is $\frac{-4}{\sqrt{-7} }$ and scalar component of u in the direction v is $\frac{-4}{7 }(i+j+k)$