Question

Let $\mathbf { u } , \mathbf { v } , \mathbf { w }$ be such that $| \mathbf { u } | = 1 , | \mathbf { v } | = 2 , | \mathbf { w } | = 3$. If the projection $\mathbf { V }$ along $\mathbf { u }$ is equal to that of $\mathbf { W }$ along $\mathbf { u }$ and $\mathbf { V }$, $\mathbf { W }$ are perpendicular to each other then $| \mathbf { u } - \mathbf { v } + \mathbf { w } |$ equals

• A. 14
• B. $\sqrt { 7 }$
• C. $\sqrt { 14 }$
• D. 2

Answer 1

Answer: (C)

$\sqrt { 14 }$

Answer 2

Without loss of generality, we can assume $\mathbf { v } = 2 \mathbf { i }$ and $\mathbf { w } = 3 \mathbf { j }$ Let $\mathbf { u } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ , $| \mathbf { u } | = 1$⇒ $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .....(i)

Projection of $\mathbf { V }$ along $\mathbf { u }$ = Projection of $\mathbf { W }$ along $\mathbf { u }$

⇒ $\mathbf { v } \cdot \mathbf { u } = \mathbf { w } \cdot \mathbf { u }$ ⇒ $2 \mathbf { i } \cdot ( x \mathbf { i } + y \mathbf { j } + 2 \mathbf { k } ) = 3 \mathbf { j } \cdot ( x \mathbf { i } + y \mathbf { j } + z \mathbf { k } )$

⇒ $2 x = 3 y$ ⇒ $3 y - 2 x = 0$

Now, $| \mathbf { u } - \mathbf { v } - \mathbf { w } | = | x \mathbf { i } + y \mathbf { j } + z \mathbf { k } - 2 \mathbf { i } + 3 \mathbf { j } |$

= $| ( x - 2 ) \mathbf { i } + ( y + 3 ) \mathbf { j } + z \mathbf { k } |$ = $\sqrt { ( x - 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + z ^ { 2 } }$

= $\sqrt { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) + 2 ( 3 y - 2 x ) + 13 } = \sqrt { 1 + 2 \times 0 + 13 } = \sqrt { 14 }$