Question Video: Finding the Magnitude of a Vector Given Graphically Mathematics • 12th Grade

Find the magnitude of the vector 𝐯 shown on the grid of unit squares.

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Video Transcript

Find the magnitude of the vector 𝐯 shown on the grid of unit squares below.

In this question, we’re given a graphical representation of a vector 𝐯 on a grid of unit squares and asked to use this to determine the magnitude of the vector. To do this, we can start by recalling that the magnitude of a vector represented graphically is the length of the line segment between its endpoints. Alternatively, we can think of this as the distance between its initial and terminal points. We can determine this length using the given diagram. We note that each square is unit length, and we travel seven squares to the right and one square down when traveling from the initial point of 𝐯 to its terminal point. This gives us a right triangle with sides of lengths seven and one and a hypotenuse of length equal to the magnitude of 𝐯.

We can now find the magnitude of 𝐯 by applying the Pythagorean theorem. We have the magnitude of 𝐯 squared is equal to seven squared plus one squared. We can evaluate this to obtain 50. We can then find the magnitude of 𝐯 by taking square roots of both sides of the equation, where we note that it is a length, so it is nonnegative. We have that the magnitude of 𝐯 is equal to the square root of 50. We can simplify this expression by noting 50 is equal to 25 times two. This means that we can take the root of each factor separately to obtain the magnitude of 𝐯 is equal to five root two.