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Newton's Second Law: Rotatory Version

Decompose the...

Question

Decompose the vector $\to b = 6^i - 3^j - 6^k$ into two vectors, $\to c$ and $\to d$ , one of which is parallel and the other perpendicular to the vector $\to c =^i +^j +^k$

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Solution

Component of $\to b$ along $\to c$ = $∣ ∣ ∣ \to b ∣ ∣ ∣ cos θ \to c$
$\begin{matrix} = \frac{∣ ∣ ∣ \to b ∣ ∣ ∣ (\to b . \to c) \to c}{∣ ∣ ∣ \to b ∣ ∣ ∣ ∣ ∣ \to c ∣ ∣} = \frac{(\to b . \to c) \to c}{{∣ ∣ \to c ∣ ∣}^{2}} \end{matrix}$
$= (\frac{6 - 3 - 6}{3}) (\to i + \to j + \to k)$
$\begin{matrix} \to d = - \to i - \to j - \to k \to b ⊥ \to c = \to b - \to d \end{matrix}$
$= 6 \to i - 3 \to j - 6 \to k + \to i + \to j + \to k$
$\to e = 7 \to i - 2 \to j - 5 \to k$
$\to b = 6 \to i - 3 \to j - 6 \to k = (- \to i - \to j - \to k) + (7 \to i - 2 \to j - 5 \to k)$
Parallel to $(- \to i - \to j - \to k)$
Perpendicular to $(7 \to i - 2 \to j - 5 \to k)$

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Newton's Second Law: Rotatory Version

Standard XII Physics

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