Given vector a = (1/7)(2i + 3j + 6k), b = (1/7)(3i - 6j + 2k), c = (1/7)(6i + 2j - 3k), i, j, k being a right handed orthogonal system

Question

Given a = (1/7)(2i + 3j + 6k), b = (1/7)(3i - 6j + 2k), c = (1/7)(6i + 2j - 3k), i, j, k being a right handed orthogonal system of unit vectors in space, show that vectors a, b, c is also another system.

Answer 1

To show that  \(\vec a,\,\vec b,\,\vec c\) is a right handed orthogonal system of unit vectors, we need to prove the following –

Let us consider each of these one at a time.

(a) Recall the magnitude of the vector  \(\text x\hat i+y\hat j+z\hat k\) is

Finally, we will find \(|\vec c|.\)

Hence, we have  \(|\vec a|=|\vec b|=|\vec c|=1\)

(b) Now, we will evaluate the vector  \(\vec a\times\vec b\) 

Recall the cross product of two vectors

Taking the scalar \(\cfrac17\) common, here, we have (a₁, a₂, a₃) = (2, 3, 6) and (b₁, b₂, b₃) = (3, –6, 2)

(c) Now, we will evaluate the vector  \(\vec b\times\vec c\)

Taking the scalar \(\cfrac17\) common, here, we have (a₁, a₂, a₃) = (3, –6, 2) and (b₁, b₂, b₃) = (6, 2, –3)

(d) Now, we will evaluate the vector  \(\vec c\times\vec a\)

Taking the scalar \(\cfrac17\) common, here, we have (a1, a2, a3) = (6, 2, –3) and (b1, b2, b3) = (2, 3, 6)

Thus,  \(\vec a,\,\vec b,\,\vec c\) is also another right handed orthogonal system of unit vectors.