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Show that if a and b are :(a). in the same direction then |a + b| = |a| +|b|.(b). in the opposite direction then |a − b| = |a| + |b|. |
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Answer» (a) \(|\vec a+\vec b|\) = |a| + |b| Taking R.H.S. square and solve \(=\sqrt{a^2+b^2+2ab\,cos\theta}\) a and b are same direction. so θ = 0° \(=\sqrt{a^2+b^2+2ab\,cos\,0^\circ}\) \(=\sqrt{a^2+b^2+2ab}\) \(=\sqrt{(a+b)^2}\) = |a| + |b| R.H.S. = L.H.S. (b) \(|\bar a-\bar b|\) = |a| + |b| Taking R.H.S. and squaring both side |a - b|2 \(=\sqrt{a^2+b^2-2ab\,cos\theta}\) a vector and \(\vec b\) are opposite direction θ = 180° \(=\sqrt{a^2+b^2-2ab\,cos\,180^\circ}\) \(=\sqrt{a^2+b^2-2ab\times-1}\) \(=\sqrt{a^2+b^2+2ab}\) \(=\sqrt{(a+b)^2}\) = |a| + |b| R.H.S. = L.H.S. |
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