$\text { Given } \mathbf { u } = ( 9,7 ) \text { and } \mathbf { v } = \langle 2,3 \} \text {, find the projection of } \mathbf { u } \text { onto } \mathbf { v }$
A) $( 27,21 )$
B) $\{ 6,9 \}$
C) $( 31,25 )$
D) $( 10,13 )$
E) $( 27,9 )$

Question

$\text { Given } \mathbf { u } = ( 9,7 ) \text { and } \mathbf { v } = \langle 2,3 \} \text {, find the projection of } \mathbf { u } \text { onto } \mathbf { v }$ 
A) $( 27,21 )$
B) $\{ 6,9 \}$
C) $( 31,25 )$
D) $( 10,13 )$
E) $( 27,9 )$

Quizplus · Accepted Answer

The projection of $\mathbf{u}$ onto $\mathbf{v}$ is given by $\text{proj}_{\mathbf{v}}\mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v}$. Calculating this gives $\left( \frac{(9 \cdot 2) + (7 \cdot 3)}{(2 \cdot 2) + (3 \cdot 3)} \right) \mathbf{v} = \left( \frac{18 + 21}{4 + 9} \right) \mathbf{v} = \left( \frac{39}{13} \right) \mathbf{v} = 3\mathbf{v} = 3\langle 2,3 \rangle = \{6,9\}$.

A) (27,21)( 27,21 )(27,21) B) {6,9}\{ 6,9 \}{6,9} C) (31,25)( 31,25 )(31,25)

A) $( 27,21 )$ B) $\{ 6,9 \}$ C) $( 31,25 )$