If θ is the angle between two vectors $\hat{i} - 2 \hat{j} + 3 \hat{k} and 3 \hat{i} - 2 \hat{j} + \hat{k}$ find sin θ.

Solution

$\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k} \vec{b} = 3 \hat{i} - 2 \hat{j} + \hat{k} \vec{a} x \vec{b} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 2 & 3 \\ 3 & - 2 & 1 \end{matrix}| = (4) \hat{i} + 8 \hat{j} + 4 \hat{k} | \vec{a} x \vec{b} | = \sqrt{(4)^{2} + (8)^{2} + (4)^{2}} = \sqrt{16 + 64 + 16} = \sqrt{96} = 4 \sqrt{6} | \vec{a} | = \sqrt{1^{2} + (- 2)^{2} + 3^{2}} = \sqrt{14} | \vec{b} | = \sqrt{3^{2} + (- 2)^{2} + 1^{2}} = \sqrt{14} ∵ | \vec{a} x \vec{b} | = | \vec{a} | | \vec{b} | \sin θ where θ is the angle between \vec{a} and \vec{b} \Rightarrow \sin θ = \frac{4 \sqrt{6}}{\sqrt{14} \sqrt{14}} = \frac{4 \sqrt{6}}{14} = \frac{2}{7} \sqrt{6} \sin θ = \frac{2}{7} \sqrt{6}$

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If θ is the angle between two vectors $\hat{i} - 2 \hat{j} + 3 \hat{k} and 3 \hat{i} - 2 \hat{j} + \hat{k}$ find sin θ.

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If θ is the angle between two vectors i^ - 2j^ + 3k^ and 3i^ - 2j^ + k^ find sin θ.

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If θ is the angle between two vectors $\hat{i} - 2 \hat{j} + 3 \hat{k} and 3 \hat{i} - 2 \hat{j} + \hat{k}$ find sin θ.

Classify the following measures as scalars and vectors.
(i) 5 seconds
(ii) 1000 cm³(iii) 10 Newton
(iv) 30 km hr
(v) 10 g/cm³(vi) 20 m/s towards north

Classify the following measures as scalars and vectors.
10 kg

Classify the following measures as scalars and vectors.
2 meters north-west

Classify the following measures as scalars and vectors.
40°

Classify the following measures as scalars and vectors.
40 watt

Classify the following measures as scalars and vectors.
10^–19 coulomb

Classify the following measures as scalars and vectors.
20 m/s²

Classify the following measures as scalars and vectors.
time period

Classify the following measures as scalars and vectors.
distance