Let a -4 -2 k- b -3 -2 -7 k- and c -2 -4 k- Find a vector d which is perpendicular to both a and b and c - d -15

The vector which is perpendicular to both a and b must be parallel to a × b. Now, a × b = î amp; ĵ amp; k̂ 1 amp; 4 amp; 2 3 amp; 2 amp; 7 = ...

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Byju's Answer

Standard XII

Mathematics

Applications Scalar Triple Product

Let a =î+4 ĵ+...

Question

Let $a =^i + 4^j + 2^k, b = 3^i - 2^j + 7^k$ and $c = 2^i -^j + 4^k$ . Find a vector d which is perpendicular to both a and b and c.d = 15

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Solution

The vector which is perpendicular to both a and b must be parallel to $a \times b$ .
Now, $a \times b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 1 & 4 & 2 3 & - 2 & 7 \end{matrix} ∣ ∣ ∣ ∣ ∣ =^i (28 + 4) -^j (7 - 6) +^k (- 2 - 12) = 32^i -^j - 14^k$
Let $d = λ (a \times b) = λ (32^i -^j - 14^k)$
Also, $c . d = 15 \Rightarrow (2^i -^j + 4^k) . λ (32^i -^j - 14^k) = 15$
$\Rightarrow 2 \times (32 λ) + (- 1) \times (- λ) + 4 \times (- 14 λ) = 15 \Rightarrow 64 λ + λ - 56 λ = 15 \Rightarrow λ = \frac{15}{9} = \frac{5}{3}$
$∴$ Required vector, $d = \frac{5}{3} (32^i -^j - 14^k)$

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Let a=^i+4^j+2^k,b=3^i−2^j+7^k and c=2^i−^j+4^k. Find a vector d which is perpendicular to both a and b and c.d = 15

Let $a =^i + 4^j + 2^k, b = 3^i - 2^j + 7^k$ and $c = 2^i -^j + 4^k$ . Find a vector d which is perpendicular to both a and b and c.d = 15