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Express the vector a-›=5i^ +2j^+5k^ as the sum …

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Express the vector a^-›=5i^{^} +2j^{^}+5k^{^} as the sum of two vectors such that one is parallel to vector b^-›=3i^{^} +k^{^} and other is perpendicular to b^-› .

Posted by Prabhnoor Singh 7 years, 3 months ago

CBSE > Class 12 > Mathematics

1 answers

Naveen Sharma 7 years, 3 months ago

Given : $\overset{⇀}{a} = 5 \hat{i} + 2 \hat{j} + 5 \hat{k}, \overset{⇀}{b} = 3 \hat{i} + \hat{k}$

Let Vector $\overset{⇀}{a} = \overset{⇀}{a_{1}} + \overset{⇀}{a_{2}}$ , Where $\overset{⇀}{a_{1}} i s p a r a l l e l t o \overset{⇀}{b} a n d \overset{⇀}{a_{2}} i s p e r p e n d i c u l a r t o \overset{⇀}{b}$ .

$a s \overset{⇀}{a_{1}} i s p a r a l l e l t o \overset{⇀}{b} \overset{⇀}{a_{1}} = λ (3 \overset{⇀}{i} + \overset{⇀}{k}) = > 3 λ \overset{⇀}{i} + λ \overset{⇀}{k} . . . . . . . . . . . . . . . . . . . (1)$

Also

$\overset{⇀}{a} = \overset{⇀}{a_{1}} + \overset{⇀}{a_{2}}$ ,

=> $\overset{⇀}{a_{2}} = \overset{⇀}{a} - \overset{⇀}{a_{1}}$ ,

=> $\overset{⇀}{a_{2}} = 5 \overset{⇀}{i} + 2 \overset{⇀}{j} + 5 \overset{⇀}{k} - 3 λ \overset{⇀}{i} - λ \overset{⇀}{k}$

=> $\overset{⇀}{a_{2}} = (5 - 3 λ) \overset{⇀}{i} + 2 \overset{⇀}{j} + (5 - λ) \overset{⇀}{k} . . . . . . . . . . . . . . . . . (2)$

As $\overset{⇀}{a_{2}} i s p e r p e n d i c u l a r t o \overset{⇀}{b}$

So, $\overset{⇀}{a_{2}} \cdot \overset{⇀}{b} = 0$

$((5 - 3 λ) \overset{⇀}{i} + 2 \overset{⇀}{j} + (5 - λ) \overset{⇀}{k}) . (3 \overset{⇀}{i} + \overset{⇀}{k}) = 0 = > 15 - 9 λ + 5 - λ = 0 = > 10 λ = 20 = > λ = 2$

Put value of $λ$ in both equations.

$\overset{⇀}{a_{1}} = 6 \overset{⇀}{i} + 2 \overset{⇀}{k}$

$\overset{⇀}{a_{2}} = - \overset{⇀}{i} - 2 \overset{⇀}{j} + 3 \overset{⇀}{k}$

ANSWER

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