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Chapter 11: Problem 6

What is the magnitude of a vector joining two points \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right) ?\)

Short Answer

Expert verified
Question: Given two points in space, P(2, 4, -1) and Q(-3, 1, 5), find the magnitude of the vector joining these points. Answer: To find the magnitude of the vector joining the points P and Q, follow the steps mentioned above. For the given points, the components are: x = -3 - 2 = -5 y = 1 - 4 = -3 z = 5 - (-1) = 6 Now, using the magnitude formula, we get: magnitude = 鈭�((-5)^2 + (-3)^2 + (6)^2) magnitude = 鈭�(25 + 9 + 36) magnitude = 鈭�(70) Therefore, the magnitude of the vector joining the points P and Q is 鈭�(70).

Step by step solution

01

Find the vector components

The components of the vector joining points \(P(x_{1}, y_{1}, z_{1})\) and \(Q(x_{2}, y_{2}, z_{2})\) are given by: $$ \begin{aligned} &x = x_{2} - x_{1},\\ &y = y_{2} - y_{1},\\ &z = z_{2} - z_{1}. \end{aligned} $$
02

Calculate the magnitude of the vector

Now that we have found the components of the vector, we can calculate its magnitude by using the formula for the magnitude of a vector in 3 dimensions: $$ \text{magnitude} = \sqrt{x^{2} + y^{2} + z^{2}} $$ Substitute the components from Step 1 into this formula to get: $$ \text{magnitude} = \sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2 + (z_{2} - z_{1})^2} $$ Now, you have the magnitude of the vector joining points \(P\) and \(Q\).

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