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What Is the Length of a Vector?

A vector a with its x and y component

The length of a vector a = (x,y) is denoted by an absolute value sign |a|, and can be found by using this formula (notice the similarity to the Pythagorean theorem):

Formula

The Length of a Vector

|a| = | (x,y)| = x2 + y2

Example 1

Find the length of the vector v = (3, 4).

|v| = | (3, 4)| = 32 + 42 = 9 + 16 = 25 = 5

|v| = | (3, 4)| = 32 + 42 = 9 + 16 = 25 = 5

Example 2

For what values of t does the vector (t,t2) have length 2?

In this case, you use the formula as an equation and solve it for t. This is how you do it:

2 = (t ) 2 + (t2 ) 2 = t2 (1 + t2 ) = t1 + t2. You divide each side of the equation by t and get

2 t = 1 + t2 Squaring 2 t2 = 1 + t2 | t2 2 = t2 + t4 Move over 0 = t4 + t2 2 u = t2 0 = u2 + u 2 Factorizing 0 = (u + 2) (u 1) .

2 t = 1 + t2 Squaring 2 t2 = 1 + t2 | t2 2 = t2 + t4 Move over 0 = t4 + t2 2 Substitute u = t2 0 = u2 + u 2 Factorizing 0 = (u + 2) (u 1) .

You can now substitute back. Then you get this from the first factor: u + 2 = 0 (1) t2 + 2 = 0 t2 = 2

From the second factor, you get u 1 = 0 (2) t2 1 = 0 t2 = 1 t = ±1.

Since t2 = 2 from (1) has no solution, t = ±1 from (2).

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