# If length of vector v= 2 and length of vector w= 3, what are the largest and smallest values for length of (vector v - vector w)? Illustrate your answers.

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You need to use the formula that helps you to evaluate the length of a vector `bar v = a bar i + b bar j` and `bar w = c bar i + d bar j` , such that:

`|bar v| = sqrt(a^2 + b^2)`

`|bar w| = sqrt(c^2 + d^2)`

The problem provides the lengths of vectors bar v and bar w such that:

`2 = sqrt(a^2 + b^2) => 4 = a^2 + b^2`

`3 = sqrt(c^2 + d^2) => 9 = c^2 + d^2`

You need to evaluate the length of the new vector `bar v - bar w` but first you need to evaluate the new vector `bar u = bar v - bar w` , such that:

`bar u = bar v - bar w => bar u = a bar i + b bar j - c bar i - d bar j => bar u = (a - c)bar i + (b - d) bar j`

Evaluating the length of the vector `bar u = bar v - bar w` yields:

`|bar u| = sqrt((a - c)^2 + (b - d)^2)`

Expanding the squares, yields:

`|bar u| = sqrt(a^2 - 2ac + c^2 + b^2 - 2bd + d^2)`

Replacing 4 for `a^2 + b^2` and 9 for` c^2 + d^2 ` yields:

`|bar u| = sqrt(4 + 9 - 2ac - 2bd) => |bar u| = sqrt(13 - 2ac - 2bd)`

`|bar u| EE<=>13 - 2ac - 2bd >= 0 =>- 2ac - 2bd >= -13 => 2(ac + bd) <= 13 =>ac + bd <= 13/2`

**Hence, evaluating the conditions concerning the length of the vector `bar u = bar v - bar w` yields **`ac + bd <= 13/2.`