Questions: Find the distance between the two points, and sketch the points and the line between them. (-4,2,1) and (3,0,-3)

Solution Steps

To find the distance between two points in 3-dimensional space, we use the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] We will plug in the coordinates of the given points into this formula to calculate the distance.

Given points: \[ \text{Point 1} = (-4, 2, 1) \] \[ \text{Point 2} = (3, 0, -3) \]

Calculate the differences in the \(x\), \(y\), and \(z\) coordinates: \[ \Delta x = 3 - (-4) = 7 \] \[ \Delta y = 0 - 2 = -2 \] \[ \Delta z = -3 - 1 = -4 \]

Use the distance formula for 3-dimensional space: \[ \text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} \] Substitute the calculated differences: \[ \text{Distance} = \sqrt{(7)^2 + (-2)^2 + (-4)^2} \] \[ \text{Distance} = \sqrt{49 + 4 + 16} \] \[ \text{Distance} = \sqrt{69} \] \[ \text{Distance} \approx 8.3066 \]

Final Answer