Video Transcript
The vector 𝐀 can be written in the form 𝑎 subscript 𝑥 times 𝐢 hat plus 𝑎 subscript 𝑦 times 𝐣 hat. What is the value of 𝑎 subscript 𝑥? What is the value of 𝑎 subscript 𝑦?
Okay, so in this question, we’re given a diagram showing a vector that’s labeled as 𝐀. We’re told that 𝐀 can be written as 𝑎 subscript 𝑥 times 𝐢 hat plus 𝑎 subscript 𝑦 times 𝐣 hat. This way of expressing a vector is known as writing it in component form. We can understand why it’s called this by recalling what 𝐢 hat and 𝐣 hat represent. 𝐢 hat is the unit vector in the 𝑥-direction, and 𝐣 hat is the unit vector in the 𝑦-direction.
We can recall that the 𝑥-direction is the horizontal direction, while the 𝑦-direction is the vertical direction. So then, on the diagram that we’re given, this horizontal axis is the 𝑥-axis and this vertical one is the 𝑦-axis. The unit vector 𝐢 hat is the vector in the 𝑥-direction with a magnitude of one. Similarly, 𝐣 hat is a vector with a magnitude of one in the 𝑦-direction. When we write the vector 𝐀 in component form, we see that it has two terms.
The first term is 𝑎 subscript 𝑥 multiplied by 𝐢 hat. And since 𝐢 hat defines the 𝑥-axis direction, then 𝑎 subscript 𝑥 is the 𝑥-component of the vector 𝐀. The second term is 𝑎 subscript 𝑦 multiplied by 𝐣 hat. And 𝐣 hat defines the 𝑦-direction, so 𝑎 subscript 𝑦 is the 𝑦-component of 𝐀.
The first part of the question is asking us to find the value of 𝑎 subscript 𝑥, so that’s the 𝑥-component of this vector 𝐀 in the diagram. To find this 𝑥-component, we need to trace vertically downward from the tip of vector 𝐀 until we get to the 𝑥-axis. Then, we need to count across the number of squares from this origin to the point where this line we’ve drawn meets the axis. Counting these squares, we find that there are one, two, three, four five, six, seven.
So, our line that we’ve drawn traced down from the tip of vector 𝐀 to the 𝑥-axis meets that axis at a value of seven. This means that the 𝑥-component of the vector 𝐀, which is our value for 𝑎 subscript 𝑥, is equal to seven.
Let’s now have a look at the second part of the question, which is asking us to find the value of 𝑎 subscript 𝑦. Now, we said earlier that just as 𝑎 subscript 𝑥 was the 𝑥-component of the vector 𝐀, 𝑎 subscript 𝑦 is the 𝑦-component of this vector. To find the 𝑥- component of 𝐀, we traced downward from the tip of the vector 𝐀 until we got to the 𝑥-axis. Similarly, to find this vector’s 𝑦-component, we need to trace across horizontally from the tip of the vector until we get to the 𝑦-axis. When we do this, we get this dashed horizontal line here.
If we count up the squares from the origin to the point where this line meets the 𝑦-axis, we find that this number of squares is equal to one, two, three, four. So, our horizontal line from the tip of vector 𝐀 meets the 𝑦-axis at a value of four. This means that the 𝑦-component of 𝐀, which is the value of 𝑎 subscript 𝑦, is equal to four.
