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Chapter 14: Problem 5

If \(A\) has coordinates \((4,3,0)\) and B has coordinates \((-2,1,9)\) find \(\overrightarrow{\mathrm{AB}}\) and \(|\overrightarrow{\mathrm{AB}}|\).

Short Answer

Expert verified
Answer: The vector AB is (-6, -2, 9), and its magnitude is 11.

Step by step solution

01

Find \(\overrightarrow{\mathrm{AB}}\)

To find the vector \(\overrightarrow{AB}\), we need to find the difference between the coordinates of point B and point A. For this, we will subtract the x, y, and z coordinates of point A from the corresponding coordinates of point B: \(\overrightarrow{\mathrm{AB}} = (B_x - A_x, B_y - A_y, B_z - A_z) = (-2 - 4, 1 - 3, 9 - 0)\) Now we just need to evaluate each difference: \(\overrightarrow{\mathrm{AB}} = (-6, -2, 9)\) So, the vector \(\overrightarrow{\mathrm{AB}}\) is \((-6, -2, 9)\).
02

Find \(|\overrightarrow{\mathrm{AB}}|\)

To find the magnitude (length) of the vector \(\overrightarrow{\mathrm{AB}}\), we can use the distance formula, given by: \(|\overrightarrow{\mathrm{AB}}| = \sqrt{(-6)^2 + (-2)^2 + 9^2}\) Now we need to evaluate the squares and the square root: \(|\overrightarrow{\mathrm{AB}}| = \sqrt{36 + 4 + 81} = \sqrt{121}\) Since the square root of 121 is 11: \(|\overrightarrow{\mathrm{AB}}| = 11\) Thus, the magnitude of vector \(\overrightarrow{\mathrm{AB}}\) is 11. In conclusion, the vector \(\overrightarrow{\mathrm{AB}}\) is \((-6, -2, 9)\), and its magnitude is 11.

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