Section 3: Subtracting Vectors
When we're subtracting vectors, we can use the same process we had with addition. Before we do that, we need to take advantage of one mathematical technicality. To illustrate, think of something like...
4 - 2... which is technically ... 4 + (-2)
We can think of subtraction as adding the opposite. This essentially means that if we subtract b from a, we can think of it as adding -b (which is called the additive inverse). This works in the world of vectors as well...
a - b = a + (-b)
4 - 2... which is technically ... 4 + (-2)
We can think of subtraction as adding the opposite. This essentially means that if we subtract b from a, we can think of it as adding -b (which is called the additive inverse). This works in the world of vectors as well...
a - b = a + (-b)
Negating a Vector
Let's talk about the vector a = [2, 3], we're looking for a vector b= -a. To draw this vector we need to negate it. To do that, multiply each coordinate by -1 (or just switch the sign on each component).
-a= -1 [2, 3] = [-2, -3] That means b = [ -2, -3]. To see how the negation affects the vector in the plane. The only thing negation will do is reflect the vector over the x axis like a mirror. Test Your Understanding Negate the following vectors. 1) u = [ 1, -3 ] 2) v = [-4, -6 ] 3) n = [ 0.5 , 0.75 ] |
Subtraction becomes Adding Again!
All we need to do is rewrite the equations as x = a + (-b) + (-c) and we'll just have vector addition all over again! The only extra step is to negate the vectors b and c before putting each of the pieces tip to tail.
Test Your Understanding
Perform each subtraction.
Perform each subtraction.