Let a- - - - - - k- b- -k- and c- - - - - k- be three vectors- A vector v- in the plane ofa- and b- whose projection on c- is 1-3- is given by

Let v⃗= a⃗ +λb⃗ v⃗=(1+ λ) î+(1 λ) ĵ+(1+λ)k̂ Projection of v⃗ on c⃗= 1/√(3) ⇒ v⃗·c⃗/|c⃗|=1/√(3) ⇒ (1+ λ) (1 λ) (1+λ)/√(3)=1/√(3) ⇒ 1+λ 1+λ 1 λ=1 ⇒λ 1=1 ⇒λ ...

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Byju's Answer

Standard XI

Mathematics

Dot Product of Two Vectors

Let a⃗ = î + ...

Question

Let $\to a =^i +^j +^k, \to b =^i -^j +^k$ and $\to c =^i -^j -^k$ be three vectors. A vector $\to v$ in the plane of
$\to a$ and $\to b$ , whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ , is given by

$^i - 3^j + 3^k$

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$- 3^i - 3^j +^k$

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$3^i -^j + 3^k$

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$^i - 3^j - 3^k$

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Solution

The correct option is C
$3^i -^j + 3^k$

Let $\to v = \to a + λ \to b \to v = (1 + λ)^i + (1 - λ)^j + (1 + λ)^k Projection of \to v on \to c = \frac{1}{\sqrt{3}} \Rightarrow \frac{\to v \cdot \to c}{| \to c |} = \frac{1}{\sqrt{3}} \Rightarrow \frac{(1 + λ) - (1 - λ) - (1 + λ)}{\sqrt{3}} = \frac{1}{\sqrt{3}}$

$\Rightarrow 1 + λ - 1 + λ - 1 - λ = 1 \Rightarrow λ - 1 = 1 \Rightarrow λ = 2 ∴ \to v = 3^i -^j + 3^k$

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Similar questions

Let $\to a =^i +^j +^k, \to b =^i -^j +^k$ and $\to c =^i -^j -^k$ be three vectors. A vector $\to v$ in the plane of
$\to a$ and $\to b$ , whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ , is given by