Sum of coefficients of i- j and k in the cross product 2i - 3j - 4k - i - j - k will be-

The two given vectors are u = 2i + 3j + 4k v = i j + k Their cross product will be given by the following determinant u×v = i amp; j amp; k nbsp; 2 ...

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Standard XIII

Mathematics

Cross Product of Two Vectors

Sum of coeffi...

Question

Sum of coefficients of $ˆ i, ˆ j$ and $ˆ k$ in the cross product $(2 ˆ i + 3 ˆ j + 4 ˆ k)$ $\times$ $(ˆ i - ˆ j + ˆ k)$ will be___.

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Solution

The two given vectors are
$\to u = 2 ˆ i + 3 ˆ j + 4 ˆ k$
$\to v = ˆ i - ˆ j + ˆ k$
Their cross product will be given by the following determinant
$\to u \times \to v = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 2 & 3 & 4 1 & - 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
The second row of the determinant contains the coefficients of $ˆ i, ˆ j, ˆ k$ in vector $\to u$ and the third row contains the corresponding coefficients in vector $\to v$ .
The determinant on evaluation will give:-
$ˆ i ((3 \times 1) - (- 1 \times 4)) . ˆ j ((2 \times 1) - (1 \times 4)) + ˆ k ((2 \times 1) - (3 \times 1))$
$= 7 ˆ i + 2 ˆ j - 5 ˆ k$
So the sum of the coefficient of $ˆ i, ˆ j$ and $ˆ k$ will be $(7 + 2 - 5) = 4.$

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3 \hat{i} + \hat{j} - 2 \hat{k} and \hat{i} - 3 \hat{j} + 4 \hat{k}

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\hat{i} - 3 \hat{j} + \hat{k} and \hat{i} + \hat{j} + \hat{k} .

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Q. Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
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\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}

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\vec{a} = 11 \hat{i}, \vec{b} = 2 \hat{j}, \vec{c} = 13 \hat{k}

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Q. Find

[\vec{a} \vec{b} \vec{c}]

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\vec{a} = 2 \hat{i} - 3 \hat{j}, \vec{b} = \hat{i} + \hat{j} - \hat{k} and \vec{c} = 3 \hat{i} - \hat{k}

(ii)

\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k} and \vec{c} = \hat{j} + \hat{k}

(iii)

\vec{a} = 2 \hat{i} + 3 \hat{j} + \hat{k}, \vec{b} = \hat{i} - 2 \hat{j} + \hat{k} and \vec{c} = - 3 \hat{i} + \hat{j} + 2 \hat{k}

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Sum of coefficients of ˆi,ˆj and ˆk in the cross product (2ˆi+3ˆj+4ˆk) × (ˆi−ˆj+ˆk) will be___.

Sum of coefficients of $ˆ i, ˆ j$ and $ˆ k$ in the cross product $(2 ˆ i + 3 ˆ j + 4 ˆ k)$ $\times$ $(ˆ i - ˆ j + ˆ k)$ will be___.