– $ \space a \space = \space (4, \space 7, \space -4), \space b \space = \space (3, \space -1, \space 1) $
The main objective of this question is to find the scalar and vector of one vector onto the other vector.
This question uses the concept of vector and scalar projection. A vector projection is indeed the vector that is made when one vector is broken up into two parts, one of which is parallel to the 2nd vector and the other of which is not while scalar projection is sometimes meant by the term scalar component.
Expert Answer
In this question, we have to find the projection of one vector on the other vector. So first, we have to find the dot product.
\[ \space a \space . \space b \space = \space (4, \space 7, \space -4) \space . \space (3, \space -1, \space 1) \]
\[ \space 4 \space . \space 3 \space + \space 7 \space . \space (-1) \space + \space (-4) \space . \space 1 \]
\[ \space = \space 12 \space – \space 7 \space – \space 4 \]
\[ \space = \space 1 \]
Now magnitude is:
\[ \space |a| \space = \space \sqrt{4^2 \space + \space 7^2 \space + \space (-4)^2} \]
\[ \space = \space \sqrt{16 \space + \space 49 \space + \space 16} \]
\[ \space = \space \sqrt{81} \]
\[ \space = \space 9 \]
Now scalar projection is:
\[ \space comp_a b \space = \space \frac{a.b}{|a|} \]
Substituting the values will result in:
\[ \space comp_a b \space = \space \frac{1}{9} \]
Now vector projection is:
\[ \space comp_a b \space = \space [comp_a b]\frac{a}{|a|} \]
By substituting values, we get:
\[ \space = \space \frac{4}{81}, \space \frac{7}{81}, \space – \frac{4}{81} \]
Numerical Answer
The scalar projection is:
\[ \space comp_a b \space = \space \frac{1}{9} \]
And the vector projection is:
\[ \space = \space \frac{4}{81}, \space \frac{7}{81}, \space – \frac{4}{81} \]
Example
Find the scalar projection of vector $ b $ on $ a $.
- $ \space a \space = \space (4, \space 7, \space -4), \space b \space = \space (3, \space -1, \space -4) $
First, we have to find the dot product.
\[ \space a \space . \space b \space = \space (4, \space 7, \space -4) \space . \space (3, \space -1, \space -4) \]
\[ \space 4 \space . \space 3 \space + \space 7 \space . \space (-1) \space + \space (-4) \space . \space -4 \]
\[ \space = \space 12 \space – \space 7 \space + \space 16 \]
\[ \space = \space 21 \]
Now magnitude is:
\[ \space |a| \space = \space \sqrt{4^2 \space + \space 7^2 \space + \space (-4)^2} \]
\[ \space = \space \sqrt{16 \space + \space 49 \space + \space 16} \]
\[ \space = \space \sqrt{81} \]
\[ \space = \space 9 \]
Now scalar projection is:
\[ \space comp_a b \space = \space \frac{a.b}{|a|} \]
Substituting the values will result in:
\[ \space comp_a b \space = \space \frac{21}{9} \]
Thus the scalar projection of vector $ b $ on $ a $ is:
\[ \space comp_a b \space = \space \frac{21}{9} \]