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Applications of Dot Product

Let a, b, c b...

Question

Let a, b, c be the vectors of lengths 3, 4, 5 respectively. If each one is perpendicular to the sum of other two, then find the magnitude of vector $a + b + c .$

3 \sqrt{3}

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4 \sqrt{2}

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5 \sqrt{2}

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6 \sqrt{2}

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Solution

The correct option is C $5 \sqrt{2}$
Given, $a . (b + c) = 0, b . (c + a) = 0, c . (a + b) = 0$
Adding these, $2 (a . b + b . c + c . a) = 0$ Also $a^{^{2}} = ∣ ∣ a^{2} ∣ ∣ = 9, b^{_{2}} = 16, c^{2} = 25$
Now ${(a + b + c)}^{2} = Σ a^{2} + 2 Σ a . b = 9 + 16 + 25 + 0 + = 50$
$∴ | a + b + c | = 5 \sqrt{2}$

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Let a, b, c be the vectors of lengths 3, 4, 5 respectively. If each one is perpendicular to the sum of other two, then find the magnitude of vector a+b+c.

The correct option is C 5√2Given, a.(b+c)=0,b.(c+a)=0,c.(a+b)=0 Adding these, 2(a.b+b.c+c.a)=0 Also a2=∣∣a2∣∣=9,b2=16,c2=25 Now (a+b+c)2=Σa2+2Σa.b=9+16+25+0+=50 ∴|a+b+c|=5√2

Let a, b, c be the vectors of lengths 3, 4, 5 respectively. If each one is perpendicular to the sum of other two, then find the magnitude of vector $a + b + c .$