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Applications of Cross Product

If | a⃗| =2...

Question

If $| \to a | = 2, ∣ ∣ \to b ∣ ∣ = 3$ and $\to a, \to b$ are mutually perpendicular, then the area of the triangle whose vertices are $\to 0, \to a + \to b, \to a - \to b$ is

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Solution

Let the position vectors of the points $A, B, C$ are $\to 0, \to a + \to b, \to a - \to b$ and $θ = 90$ .
$∴$ Area of triangle $= \frac{1}{2} ∣ ∣ \to A B \times \to A C ∣ ∣$
$= \frac{1}{2} ∣ ∣ (\to a + \to b) \times (\to a - \to b) ∣ ∣$
$= \frac{1}{2} ∣ ∣ 2 \to b \times \to a ∣ ∣$
$= ∣ ∣ \to b ∣ ∣ | \to a | sin θ$
$= 3 \times 2 sin 90$
$= 6$

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(\to a + \to b) \cdot (\to a + \to b) = {| \to a |}^{2} + {∣ ∣ \to b ∣ ∣}^{2}

\to a \to b

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∣ ∣ \to a \times \to b ∣ ∣ + ∣ ∣ \to b \times \to c ∣ ∣ + | \to c \times \to a | =

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| \to a \times \to b | + | \to b \times \to c | + | \to c \times \to a | =

Q. Magnitudes of vectors

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3, 4, 5

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