JEE MAIN - Mathematics (2021 - 26th August Evening Shift - No. 19)
If the projection of the vector $$\widehat i + 2\widehat j + \widehat k$$ on the sum of the two vectors $$2\widehat i + 4\widehat j - 5\widehat k$$ and $$ - \lambda \widehat i + 2\widehat j + 3\widehat k$$ is 1, then $$\lambda$$ is equal to __________.
Answer
5
Explanation
$$\overrightarrow a = \widehat i + 2\widehat j + \widehat k$$
$$\overrightarrow b = (2 - \lambda )\widehat i + 6\widehat j - 2\widehat k$$
$${{\overrightarrow a \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = 1,\overrightarrow a \,.\,\overrightarrow b = 12 - \lambda $$
$$\left( {\overrightarrow a \,.\,\overrightarrow b } \right) = |\overrightarrow b {|^2}$$
$$\lambda$$2 $$-$$ 24$$\lambda$$ + 144 = $$\lambda$$2 $$-$$ 4$$\lambda$$ + 4 + 40
20$$\lambda$$ = 100 $$\Rightarrow$$ $$\lambda$$ = 5
$$\overrightarrow b = (2 - \lambda )\widehat i + 6\widehat j - 2\widehat k$$
$${{\overrightarrow a \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = 1,\overrightarrow a \,.\,\overrightarrow b = 12 - \lambda $$
$$\left( {\overrightarrow a \,.\,\overrightarrow b } \right) = |\overrightarrow b {|^2}$$
$$\lambda$$2 $$-$$ 24$$\lambda$$ + 144 = $$\lambda$$2 $$-$$ 4$$\lambda$$ + 4 + 40
20$$\lambda$$ = 100 $$\Rightarrow$$ $$\lambda$$ = 5
Comments (0)
