Question

Find the projection of the vector A = î - 2ġ + k on the vector B = 4 i - 4ſ + 7k. 15. Given the vectors A = 2 i +3 ſ +6k and B = i +59 +3k. How much of vector B is along vector A?

Answer 1

Part 1)

Projection of vector A on vector B equals 19 units

Part 2)

Projection of vector B' on vector A' equals 35 units

Explanation:

For 2 vectors A and B the projection of A on B is given by the vector dot product of vector A and B

Given

`\overrightarrow{v_(a)}=\widehat{i}-2\widehat{j}+\widehat{k}`

Similarly vector B is written as

`\overrightarrow{v_(b)}=4\widehat{i}-4\widehat{j}+7\widehat{k}`

Thus the vector dot product of the 2 vectors is obtained as

`\overrightarrow{v_(a)}\cdot \overrightarrow{v_(b)}=(\widehat{i}-2\widehat{j}+\widehat{k})\cdot (4\widehat{i}-4\widehat{j}+7\widehat{k})\\\\\overrightarrow{v_(a)}\cdot \overrightarrow{v_(b)}=1\cdot 4+2\cdot 4+1\cdot 7=19`

Part 2)

Given vector A' as

`\overrightarrow{v_(a')}=2\widehat{i}+3\widehat{j}+6\widehat{k}`

Similarly vector B' is written as

`\overrightarrow{v_(b')}=\widehat{i}+5\widehat{j}+3\widehat{k}`

Thus the vector dot product of the 2 vectors is obtained as

`\overrightarrow{v_(b')}\cdot \overrightarrow{v_(a')}=(\widehat{i}+5\widehat{j}+3\widehat{k})\cdot (2\widehat{i}+3\widehat{j}+6\widehat{k})\\\\\overrightarrow{v_(a')}\cdot \overrightarrow{v_(b')}=1\cdot 2+5\cdot 3+3\cdot 6=35`