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

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Given  \(\vec a=\hat i+4\hat j + 2\hat k,\) \(\vec b=3\hat i-2\hat j+7\hat k\)  and  \(\vec c=2\hat i-\hat j+4\hat k\)

We need to find a vector \(\vec d\) perpendicular to  \(\vec a\) and  \(\vec b\) such that \(\vec c.\vec d=15.\)

Recall a vector that is perpendicular to two vectors

Here, we have (a1, a2, a3) = (1, 4, 2) and (b1, b2, b3) = (3, –2, 7)

So,  \(\vec d\) is a vector parallel to  \(\vec a\times\vec b\).

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