1
Question
If →α=2^i+3^j−^k,→β=−^i+2^j−4^k,→γ=^i+^j+^k, then what is the value of (→α×→β)⋅(→α×→γ)?
If →α=2^i+3^j−^k,→β=−^i+2^j−4^k,→γ=^i+^j+^k, then what is the value of (→α×→β)⋅(→α×→γ)?
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Solution
The correct option is A −74
→α=2^i+3^j−^k
→β=−^i+2^j−4^k
→γ=^i+^j+^k
→α×→β=∣∣
∣
∣∣^i^j^k23−1−12−4∣∣
∣
∣∣
→α×→β=^i[−12+2]−^j[−8−1]+^k[4+3]
→α×→β=−10^i+9^j+7^k⟶1
→α×→γ=∣∣
∣
∣∣^i^j^k23−1111∣∣
∣
∣∣
→α×→γ=^i[3+1]−^j[2+1]+^k[2−3]
→α×→γ=4^i−3^j−^k⟶2
Now, (→α×→β)⋅(→α×→γ)=(−10^i+9^j+7^k)⋅(4^i−3^j−^k)
=(−10)(4)+9(−3)+7(−1)
=−40−27−7
=−74
→α=2^i+3^j−^k
→β=−^i+2^j−4^k
→γ=^i+^j+^k
→α×→β=∣∣ ∣ ∣∣^i^j^k23−1−12−4∣∣ ∣ ∣∣
→α×→β=^i[−12+2]−^j[−8−1]+^k[4+3]
→α×→β=−10^i+9^j+7^k⟶1
→α×→γ=∣∣ ∣ ∣∣^i^j^k23−1111∣∣ ∣ ∣∣
→α×→γ=^i[3+1]−^j[2+1]+^k[2−3]
→α×→γ=4^i−3^j−^k⟶2
Now, (→α×→β)⋅(→α×→γ)=(−10^i+9^j+7^k)⋅(4^i−3^j−^k)
=(−10)(4)+9(−3)+7(−1)
=−40−27−7
=−74
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