Let a -2 -k- and b -2 -k- be two vectors- Consider a vector c - a - b - - - R- If the projection of c on thevector a - b is 3 -2- then the minimum value of c - a - b - c equals

Let c=αa+βb=(2α +β)î+(α +2β)ĵ+(β α)k̂ Projection c on vector a+b =c.(a+b)/|a+b|=3√(2) ⇒((2α +β)î+(α +2β)ĵ+(β α)k̂).(3î+3 ĵ+0k̂)/|3î+3 ĵ+0k̂|=3√(2) ⇒ 9(α +β)=3√( ...

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Standard XII

Mathematics

Addition of Vectors

Let a =2 î+ĵ-...

Question

Let $\to a = 2^i +^j -^k$ and $\to b =^i + 2^j +^k$ be two vectors. Consider a vector $\to c = α \to a + β \to b, α, β \in R .$ If the projection of $\to c$ on the vector $(\to a + \to b)$ is $3 \sqrt{2},$ then the minimum value of $(\to c - (\to a \times \to b)) . \to c$ equals

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Solution

Let $\to c = α \to a + β \to b = (2 α + β)^i + (α + 2 β)^j + (β - α)^k$
Projection $\to c$ on vector $\to a + \to b$
$= \frac{\to c . (\to a + \to b)}{| \to a + \to b |} = 3 \sqrt{2}$
$\Rightarrow \frac{((2 α + β)^i + (α + 2 β)^j + (β - α)^k) . (3^i + 3^j + 0^k)}{| 3^i + 3^j + 0^k |} = 3 \sqrt{2}$
$\Rightarrow 9 (α + β) = 3 \sqrt{2} \times 3 \sqrt{2}$
$\Rightarrow α + β = 2... (i)$
$(\to a \times \to b) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & - 1 1 & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 3^i - 3^j + 3^k$
$(\to c - \to a \times \to b) . \to c = | \to c |^{2} - [\begin{matrix} \to a & \to b & \to c \end{matrix}]$
$∵ [\begin{matrix} \to a & \to b & \to c \end{matrix}] = 0$
$| c |^{2} = (2 α + β)^{2} + (α + 2 β)^{2} + (β - α)^{2} = 6 (α^{2} + β^{2} + α . β)$
For minimum value $\Rightarrow α = β = 1$
Hence, $(\to c - \to a \times \to b) . \to c$ =18

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Addition of Vectors

Standard XII Mathematics

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Let →a=2^i+^j−^k and →b=^i+2^j+^k be two vectors. Consider a vector →c=α→a+β→b, α,β∈R. If the projection of →c on the vector (→a+→b) is 3√2, then the minimum value of (→c−(→a×→b)).→c equals

Let $\to a = 2^i +^j -^k$ and $\to b =^i + 2^j +^k$ be two vectors. Consider a vector $\to c = α \to a + β \to b, α, β \in R .$ If the projection of $\to c$ on the vector $(\to a + \to b)$ is $3 \sqrt{2},$ then the minimum value of $(\to c - (\to a \times \to b)) . \to c$ equals