If a - 2 - - 2 - - 3 k- b - - - - 2 - - k- and c - 3 - - - are such that a - -b is perpedicular to c- then find the value of -

Given vectors are nbsp;a = 2 î + 2 ĵ + 3 k̂, b = î + 2 ĵ + k̂ and c = 3 î + ĵ (a + λb) = (2 î +2 ĵ + 3 k̂) + λ ( î + 2 ĵ + k̂) (a + λb = (2 λ) î + (2 + 2 ...

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Standard IX

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Progress in Mathematics

If a = 2 î + ...

Question

If $\to a = 2^i + 2^j + 3^k, \to b = -^i + 2^j +^k$ and $\to c = 3^i +^j$ are such that $\to a + λ \to b$ is perpedicular to $\to c$ , then find the value of $λ$ .

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Solution

Given vectors are $\to a = 2^i + 2^j + 3^k, \to b = -^i + 2^j +^k$ and $\to c = 3^i +^j$
$(\to a + λ \to b) = (2^i + 2^j + 3^k) + λ (-^i + 2^j +^k)$
$(\to a + λ \to b = (2 - λ)^i + (2 + 2 λ)^j + (3 + λ)^k$
Since, $(\to a + λ \to b)$ is perpedicular to $\to c$ .
Therefore, $(\to a + λ \to b) . \to c = 0$
$[(2 - λ)^i + (2 + 2 λ)^j + (3 + λ)^k] . (3^i +^j + 0^k) = 0$
$(2 - λ) .3 + (2 + 2 λ) .1 + (3 + λ) .0 = 0$
$6 - 3 λ + 2 + 2 λ = 0$
$8 - λ = 0$
$λ = 8$

Final answer:
Hence, $λ = 8$

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