Problem on the length of a vector

Consider a pyramid with square base formed by the points $(1,1,0), (1,-1, 0), (-1, 1, 0), (-1, -1, 0),$ and $(0,0,1)$. What is the length of each edge connecting the base to the apex?

Solution
To find the length of an edge, we represent it as a vector and use the formula for the length of a vector.
Recall that
Subtraction gives the vector between two points

The vector from $\mathbf{x}$ to $\mathbf{y}$ is given by $\mathbf{y} - \mathbf{x}$.
Thus, the vector starting at $(1,1,0)$ and ending at $(0,0,1)$ is given by $$(0,0,1) - (1,1,0) = (-1, -1, 1)$$
Recall that
Definition of vector length

The length of the vector $\mathbf{x} = \langle x_1, x_2, x_3 \rangle$ is $|\mathbf{x}| = \sqrt{x_1^2 + x_2^2 + x_3^2}.$
Hence, the edge going from $(1,1,0)$ to $(0,0,1)$ has length $$\sqrt{(-1)^2 + (-1)^2 + 1^2} = \sqrt{3}.$$
The lengths of the 3 other edges connecting the base to the apex are the same because of a similar calculation.

Problem on the length of a vector

Solution

Related topics