**Example 2: **

Find if the vectors

**i)** $(3, 1, -1)$ and $(3, 1, 10)$

**ii)** $(3, 4)$ and $(1, 2)$

are orthogonal to each other.

**Solution:**

**i)** Let $u$ = $(3, 1, -1)$ and $v$ = $(3, 1, 10)$

Dot product

$u \cdot v$ = $3 \times 3 + 1 \times 1 + (-1) \times 10$

= $9 + 1 - 10$

= $10 - 10$

= $0$

Since, the dot product of given vectors is zero. So, the given vectors are orthogonal vectors.

**ii)** Let $u$ = $(3, 4)$ and $v$ = $(1, 2)$

Dot product

= $3 \times 1 + 4 \times 2$

= $3 + 8$

= $11$

Since, the dot product of given vectors is not equal to zero. So, the given vectors are not orthogonal.

**Example 3: **

If the vectors $i - j + 2k$ and $2i + 2yj − 4k$ are orthogonal, then find the value of $y$.

**Solution: **

Let $u$ = $i - j + 2k$ and $v$ = $2i + 2yj − 4k$

If the given vectors are orthogonal vectors, then their dot product must be zero. That is

$u \cdot v$ = $0$

$1 \times 2 + (-1) \times 2y + 2 \times (- 4)$ = $0$

$2 - 2y - 8$ = $0$

$- 6 - 2 y$ = $0$

adding $6$ on both sides, we get

$- 6 - 2y + 6$ = $6$

$- 2y$ = $6$

Dividing by $-2$ on both sides, we get

$\frac{-2y}{-2}$ = $\frac{6}{-2}$

$Y$ = $-3$.

**Example 4: **

Determine the angle between the vectors $u$ = $(1, 3, 5)$ and $v$ = $(-2, 4, 1)$?

**Solution:**

$u \cdot v$ = $1 \times -2 + 3 \times 4 + 5 \times 1$

= $-2 + 12 + 5$

= $15$

$|u|$ = $\sqrt{1^2 + 3^2 + 5^2}$

= $\sqrt{1 + 9 + 25}$

= $\sqrt{35}$

$|v|$ = $\sqrt{- 2^2 + 4^2 + 1^2}$

= $\sqrt{4 + 16 + 1}$

= $\sqrt{21}$

$cos \theta$ = $(\frac{u \cdot v}{|u||v|})$

$cos \theta$ = $\frac{15}{\sqrt{35} x \sqrt{21}}$

= $0.553$

**Example 5: **

If $u$ = $(1, -1, 2)$ and $v$ = $(2, 3, -1)$ and compute $u x v$?

**Solution:**

$u\ x\ v$ = $\begin{vmatrix}i & j & k \\ 1 & -1 & 2\\ 2 & 3 & -1\end{vmatrix}$

= $i(1 - 6) - j(-1 -4) + k (3 + 2)$

= $-5i + 5j + 5k$