Question

How do you find two unit vectors orthogonal to both $\hat i - \hat j + \hat k$ and $4\hat j + 4\hat k$ ?

Answer 1

A vector that is perpendicular to the two vectors is said to be orthogonal to the two vectors. We know that on doing the cross product of two vectors, we get a vector that is perpendicular to both the vectors, this way we will find the vector perpendicular to the given two vectors but in this question, we have to find the unit vectors orthogonal to the given two vectors so we will find the magnitude of the obtained vector and then divide it by the magnitude.

Complete step by step answer:
Let $\overrightarrow a = \hat i - \hat j + \hat k$ and $\overrightarrow b = 4\hat j + 4\hat k$
Cross product of these two vectors is –
$\overrightarrow a \times \overrightarrow b = \left| \hat i\,\,\,\hat j\,\,\,\hat k \\\ 1\, - 1\,\,\,1 \\\ 0\,\,\,4\,\,\,4 \\\ \right| = \hat i( - 4 - 4) + \hat j(4 - 0) + \hat k(4 - 0) \\\ \Rightarrow \overrightarrow a \times \overrightarrow b = - 8\hat i + 4\hat j + 4\hat k \\\$
Now,
$\left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {{{( - 8)}^2} + {{(4)}^2} + {{(4)}^2}} \\\ \Rightarrow \left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {64 + 16 + 16} = \sqrt {96} \\\ \Rightarrow \left| {\overrightarrow a \times \overrightarrow b } \right| = 4\sqrt 6 \\\$
Unit vectors orthogonal to given two vectors are $\pm \dfrac{{\overrightarrow a \times \overrightarrow b }}{{\left| {\overrightarrow a \times \overrightarrow b } \right|}}$
$\Rightarrow \pm \dfrac{{ - 8\hat i + 4\hat j + 4\hat k}}{{4\sqrt 6 }} \\\ \Rightarrow \pm \dfrac{1}{{\sqrt 6 }}( - 2\hat i + \hat j + \hat k) \\\$
Hence, the two unit vectors orthogonal to both $\hat i - \hat j + \hat k$ and $4\hat j + 4\hat k$ are
$\dfrac{1}{{\sqrt 6 }}( - 2\hat i + \hat j + \hat k)$ and $- \dfrac{1}{{\sqrt 6 }}( - 2\hat i + \hat j + \hat k)$ .

Note: The quantities that have direction along with magnitude are known as vector quantities. Scalar quantities are the quantities having only magnitude and no direction. Even if a quantity has both direction and magnitude, it must obey certain rules to qualify as a vector. Some of the rules are vector addition, subtraction, multiplication by a scalar, scalar multiplication, vector multiplication, and differentiation. The cross product of two vectors is also a vector and is defined as the binary operation on two vectors in three-dimensional space. The obtained two orthogonal vectors lie on the same line and appear to emerge from the point of intersection of the two given vectors and go on in opposite directions as their signs are opposite.