Question

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$\left(0, 1, 5, 7,2, 1\right)$

Short answer

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

(b) The distance between the points is, $\left(-1, 5, -3, 2, 4\right), \left(2, 6, 2, 7, 6\right)$d=8.

(c) The distance between the points is,$\left(5, -2, 3, 3, 1, 0\right), \left(0, 1, 5, 7, 2, 1\right)$.$d=2\sqrt{14}$

Step-by-step solution

 (a)To find the distance between 4, -1, 2, 7  and 2, 3, 1, 9 

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 between-1, 5, -3, 2,4   and 2, 6, 2, 7,6 

(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

: (c)To find the distance between 5, -2, 3, 3,1,0and0, 1, 5, 7,2, 1

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=$2\sqrt{14}$