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Integration Using Substitution

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Question

If $\to a$ and $\to b$ are vectors in space given by $\to a = \frac{^i - 2^j}{\sqrt{5}}$ and $\to b = \frac{2^i +^j + 3^k}{\sqrt{14}}$ , then find the value of $(2 \to a + \to b) \cdot [(\to a \times \to b) \times (\to a - 2 \to b)]$ .

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Solution

$\begin{matrix} \to a \cdot \to b = \frac{1}{\sqrt{5}} \times \frac{2}{\sqrt{14}} + (\frac{- 2}{\sqrt{5}}) \times \frac{1}{\sqrt{14}} = \frac{2}{\sqrt{70}} - \frac{2}{\sqrt{70}} = 0 | \to a | = \sqrt{{(\frac{1}{\sqrt{5}})}^{2} + {(\frac{- 2}{\sqrt{5}})}^{2}} = \sqrt{\frac{1 + 4}{5}} = 1 ∣ ∣ \to b ∣ ∣ = \sqrt{{(\frac{2}{\sqrt{14}})}^{2} + {(\frac{1}{\sqrt{14}})}^{2} + {(\frac{3}{\sqrt{14}})}^{2}} = \sqrt{\frac{4 + 1 + 9}{14}} = 1 (2 \to a + \to b) \cdot [(\to a \times \to b) \times (\to a - 2 \to b)] = - (2 \to a + \to b) \cdot [(\to a - 2 \to b) \times (\to a \times \to b)] (\to a \times \to b = - \to b \times \to a) = - (2 \to a + \to b) \cdot [{(\to a - 2 \to b) . \to b} \to a - {(\to a - 2 \to b) \cdot \to a} \to b] = - (2 \to a + \to b) \cdot [(\to a \cdot \to b - 2 \to b \cdot \to b) \to a - (\to a \cdot \to a - 2 \to b \cdot \to a) \to b] = - (2 \to a + \to b) \cdot [- 2 {∣ ∣ \to b ∣ ∣}^{2} \to a - ({| \to a |}^{2}) \to b] (\to b \cdot \to b = {∣ ∣ \to b ∣ ∣}^{2}, \to a \cdot \to b = \to b \cdot \to a) = - (2 \to a + \to b) \cdot [- 2 \to a - \to b] = (2 \to a + \to b) \cdot (2 \to a + \to b) = 4 (\to a \cdot \to a) + \to b \cdot \to b + 4 \to a \cdot \to b = 4 \times 1 + 1 + 0 = 5 \end{matrix}$

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Similar questions

\to a

\to b

are vectors in space given by

\to a = \frac{^i - 2^j}{\sqrt{5}}

\to b = \frac{2^i +^j + 3^k}{\sqrt{14}}

, then the value of

(2 \to a + \to b) . [(\to a \times \to b) \times (\to a - 2 \to b)]

\to A = 2^i +^j -^k

\to B = \sqrt{2} (^i +^j)

\to A . \to B

. Hence find the angle between

\to A

\to B

. Also find the component of

\to A

\to B .

\to a

\to b

are vectors such that

| \to a + \to b | = \sqrt{29}

\to a \times (2^i + 3^j + 4^k) = (2^i + 3^j + 4^k) \times \to b

then a possible value of

(\to a + \to b) \cdot (- 7^i + 2^j + 3^k)

\to a

\to b

are vector is space given by

\to a = \frac{^i - 2^j}{\sqrt{5}}

\to b = \frac{2^i +^j + 3^k}{\sqrt{14}},

then the value of

(2 \to a + \to b) . [(\to a \times \to b) \times (\to a - 2 \to b)]

\to A

\to B

are two vectors given by

\to A = 2^i + 3^j

\to B =^i +^j

. The magnitude of the component of

\to A

\to B

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Integration Using Substitution

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