Section 2: Adding Vectors
Adding Vectors Using the Tip-to-Tail Method
When we were adding and subtracting normal numbers (scalars), it was easy enough. We didn't have to worry about directions...until now. Consider two vectors a and b that we're trying to add... a + b. We can add vectors in a plane graphically or we can just use matrix arithmetic. First, let's just place the two random vectors somewhere in space and add them up. In the animation below, the red vector is a + b.
When we were adding and subtracting normal numbers (scalars), it was easy enough. We didn't have to worry about directions...until now. Consider two vectors a and b that we're trying to add... a + b. We can add vectors in a plane graphically or we can just use matrix arithmetic. First, let's just place the two random vectors somewhere in space and add them up. In the animation below, the red vector is a + b.
A vector has a tail, where the vector begins, and a tail, where the vector ends. To add vectors together, we need to rearrange them so that they are tip to tail. After using the ability to move vectors around through space and putting them tip to tail, then you can draw a vector starting at the tail of the first vector to the tip of the last. In the pictures below, we constructed a + b and b + a. Just like regular addition, it doesn't matter which order we add them together.
Take these three vectors for example, all we need to do is arrange them tip to tail and we'll get the sum of all three.
Take these three vectors for example, all we need to do is arrange them tip to tail and we'll get the sum of all three.
Adding Multiple Vectors Tip to Tail
We can add piece by piece as shown. In the animation, we split the addition into two parts, a+b and c+d, and then add those two vectors together. On the other hand, we could arrange each piece tip to tail and draw a vector from the tail of a to the tip of d, this vector will be x also.
Parallelogram Law
While the Tip-to-Tail is reliable, there is another way to graphically add vectors together. This rule is commonly called the Parallelogram Law. The idea is to create a parallelogram out the vectors and draw a vector that divides the shape in half beginning from where the two tails of the vectors meet.
The idea here is completely opposite to the Tip-to-Tail method. Instead of moving the vectors so they're connected like a train (tip to tail), we move them so they share a common starting point. From there, complete the parallelogram by adding the other vector onto the other like you're doing the Tip-to-Tail method. In a sense, the Parallelogram Law is just an extension of the Tip-to-Tail.
Test Your Understanding
Find each sum.
Find each sum.