JEE MAIN - Mathematics (2004 - No. 26)
Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ be such that $$\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.$$ If the projection $${\overrightarrow v }$$ along $${\overrightarrow u }$$ is equal to that of $${\overrightarrow w }$$ along $${\overrightarrow u }$$ and $${\overrightarrow v },$$ $${\overrightarrow w }$$ are perpendicular to each other then $$\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right|$$ equals :
$$14$$
$${\sqrt {7} }$$
$${\sqrt {14} }$$
$$2$$
Explanation
Projection of $$\overrightarrow v $$ along $$\overrightarrow u = {{\overrightarrow v .\overrightarrow u } \over {\left| {\overrightarrow u } \right|}} = {{\overrightarrow v .\overrightarrow u } \over 2}$$
projection of $$\overrightarrow w $$ along $$\overrightarrow u = {{\overrightarrow w .\overrightarrow u } \over {\left| {\overrightarrow u } \right|}} = {{\overrightarrow w .\overrightarrow u } \over 2}$$
Given $${{\overrightarrow v .\overrightarrow u } \over 2} = {{\overrightarrow w .\overrightarrow u } \over 2}\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
Also, $$\overrightarrow v .\overrightarrow w = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$
Now $${\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right|^2}$$
$$ = {\left| {\overrightarrow u } \right|^2} + {\left| {\overrightarrow v } \right|^2} + {\left| {\overrightarrow w } \right|^2} - $$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\overrightarrow u .\overrightarrow v - 2\overrightarrow v .\overrightarrow w + 2\overrightarrow u .\overrightarrow w $$
$$ = 1 + 4 + 9 + 0$$ [ From $$(1)$$ and $$(2)$$ ] $$=14$$
$$\therefore$$ $$\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right| = \sqrt {14} $$
projection of $$\overrightarrow w $$ along $$\overrightarrow u = {{\overrightarrow w .\overrightarrow u } \over {\left| {\overrightarrow u } \right|}} = {{\overrightarrow w .\overrightarrow u } \over 2}$$
Given $${{\overrightarrow v .\overrightarrow u } \over 2} = {{\overrightarrow w .\overrightarrow u } \over 2}\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
Also, $$\overrightarrow v .\overrightarrow w = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$
Now $${\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right|^2}$$
$$ = {\left| {\overrightarrow u } \right|^2} + {\left| {\overrightarrow v } \right|^2} + {\left| {\overrightarrow w } \right|^2} - $$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\overrightarrow u .\overrightarrow v - 2\overrightarrow v .\overrightarrow w + 2\overrightarrow u .\overrightarrow w $$
$$ = 1 + 4 + 9 + 0$$ [ From $$(1)$$ and $$(2)$$ ] $$=14$$
$$\therefore$$ $$\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right| = \sqrt {14} $$
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