Question: Find the distance between the points

a)$\left(4, -1, 2, 7 \right)$and $\left(2, 3, 1, 9 \right)$

b)$\left(-1, 5, -3, 2,4 \right)$and $\left(2, 6, 2, 7,6\right)$

c)$\left(5, -2, 3, 3,1,0\right)$and$\left(0, 1, 5, 7,2, 1\right)$

Let the given points be $\left(x_1, y_1, z_1, w_1\right)=\left(4, -1, 2, 7\right), \left(x_2, y_2, z_2, w_2\right)=\left(2, 3, 1, 9\right)$.

We need to find the distance, say, , between these two points.

Then we will calculate the distance between points using distance formula.

$$\begin{gathered} d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2+{\left(w_2-w_1\right)}^2} \\ =\sqrt{{\left(2-4\right)}^2+{\left(3+1\right)}^2+{\left(1-2\right)}^2+{\left(9-7\right)}^2} \\ =\sqrt{{\left(-2\right)}^2+{\left(4\right)}^2+{\left(-1\right)}^2+{\left(2\right)}^2} \\ =\sqrt{4+16+1+4} \\ =\sqrt{25} \\ =5 \end{gathered}$$

Therefore d=5

Hence, the distance between the points $\left(4, -1, 2, 7\right), \left(2, 3, 1, 9\right)$is d=5

(b)To find the distance betweenand

Let the given points be

We need to find the distance, say, between these two points.

Then we will calculate the distance between points using distance formula

Let the given points be

$\left(x_1, y_1, z_1, u_1, v_1,w_1\right)=\left(5, -2, 3, 3, 1, 0\right), \left(x_2, y_2, z_2, u_2, v_2,w_2\right)=\left(0, 1, 5, 7, 2, 1\right)$

We need to find the distance, say,d , between these two points.

Then we will calculate the distance between points using distance formula.

$$\begin{gathered} d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2+{\left(u_2-u_1\right)}^2+{\left(v_2-v_1\right)}^2+{\left(w_2-w_1\right)}^2} \\ =\sqrt{{\left(0-5\right)}^2+{\left(1+2\right)}^2+{\left(5-3\right)}^2+{\left(7-3\right)}^2+{\left(2-1\right)}^2+{\left(1-0\right)}^2} \\ =\sqrt{{\left(-5\right)}^2+{\left(3\right)}^2+{\left(2\right)}^2+{\left(4\right)}^2+{\left(1\right)}^2+{\left(1\right)}^2} \\ =\sqrt{25+9+4+16+1+1} \\ =\sqrt{56} \\ =2\sqrt{14} \end{gathered}$$

Therefore, .d=$2\sqrt{14}$