The given figure shows a vector in a plane. Express this vector in terms of the unit vectors 𝑖 and 𝑗.
𝑖 and 𝑗 are both unit vectors. So the magnitude of 𝑖 and the magnitude of 𝑗 are both one. But the vector quantities have both magnitude and direction. So we need to know also which directions 𝑖 and 𝑗 point in.
By definition, 𝑖 is the unit vector in the 𝑥-direction. We can draw it wherever we’d like to in the plane. I’ve chosen the second quadrant. And by definition, 𝑗 is the unit vector in the 𝑦-direction.
So here’s one copy of 𝑗 in the third quadrant. We have a vector. Let’s call it 𝑣. And our task is to write this vector 𝑣 in terms of 𝑖 and 𝑗. What does this mean?
Recall that we can add two vectors together by placing them nose-to-tail. If I add 𝑖 to itself, I get the vector pointing in the 𝑥-direction with a magnitude of two. This is 𝑖 plus 𝑖 or two 𝑖. Similarly, if I put a three copies of 𝑗 together nose-to-tail, I get three 𝑗.
Now we’re ready to tackle our vector 𝑣. We can get from the initial point of 𝑣 to the terminal point of 𝑣 by first going two units right and then going 10 units up. The orange and purple vectors are nose-to-tail or, more formally, the terminal point of the orange vector is the initial point of the purple vector.
And so our green vector 𝑣 is equal to the orange vector plus the purple vector. We recognize the orange vector. It has a magnitude of two and is pointing in the 𝑥-direction. And so it is two 𝑖. How about the purple vector which points in the 𝑦-direction and has a magnitude 10? Well, this is just 10𝑗.
Our vector 𝑣, therefore, is two 𝑖 plus 10𝑗. We’ve succeeded in writing this vector in terms of the unit vectors 𝑖 and 𝑗.