Video: Finding the Magnitude of a Vector between Two Points

What is the magnitude of the vector 𝐀𝐁 where 𝐴 = (11, 3) and 𝐡 = (7, 3)?

02:24

Video Transcript

What is the magnitude of the vector 𝐀𝐁 where 𝐴 equals 11, three and 𝐡 equals seven, three?

The magnitude of a vector is its size or length. So in this case, we need to find the distance or length between point 𝐴 and point 𝐡. There are several ways of approaching this problem, we will look at two of them. Our first method will be graphically, and we will begin by plotting the two coordinates. Point 𝐴 has coordinates 11, three. Point 𝐡 has coordinates seven, three. As both points have the same 𝑦-coordinate, the distance from 𝐴 to 𝐡 will be a horizontal distance. To get from 11 to seven, we need to subtract four. As the magnitude of any vector must be positive, then the magnitude of 𝐀𝐁 is equal to four.

We could also have calculated the distance between point 𝐴 and point 𝐡 using one of our coordinate geometry formulas. The distance between any two points is equal to the square root of π‘₯ one minus π‘₯ two squared plus 𝑦 one minus 𝑦 two squared, where our two points have coordinates π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two. Substituting in our values gives us 𝑑 is equal to the square root of 11 minus seven squared plus three minus three squared.

It doesn’t matter which coordinate is π‘₯ one, 𝑦 one and which one is π‘₯ two, 𝑦 two. 11 minus seven is equal to four, and three minus three is zero. As zero squared is equal to zero, 𝑑 is equal to the square root of four squared. As our distance must be positive, this is equal to four. Once again, we have calculated that the magnitude of the vector 𝐀𝐁 is four.

Our final question will involve finding the magnitude of two separate vectors and their sum.

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