Question Video: Writing a Vector in Component Form Physics • First Year of Secondary School

Video Transcript

Write 𝐀 in component form.

Okay, so in this question, we’ve been given a diagram that shows a vector represented by a blue arrow and labeled as 𝐀. We are being asked to write this vector 𝐀 in component form. We can recall that writing a vector in component form means writing it out with an expression like this. So that’s an 𝑥-component, which we’ve labeled as 𝑎 subscript 𝑥, multiplied by 𝐢 hat, which is the unit vector in the 𝑥-direction, plus a 𝑦-component, which we’ve labeled as 𝑎 subscript 𝑦, multiplied by 𝐣 hat, the unit vector in the 𝑦-direction.

The 𝑥-direction is the horizontal one. And so on our diagram, the horizontal axis will be the 𝑥-axis. Meanwhile, the 𝑦-direction is vertical, and so the vertical axis is the 𝑦-axis.

A unit vector is a vector with a magnitude or length of one. So then, 𝐢 hat is a vector which points in the positive 𝑥-direction with a magnitude of one, so that’s one square in our diagram. Likewise, 𝐣 hat is a vector with a magnitude of one, so that’s a length of one square that points in the positive 𝑦-direction.

With this understanding of the unit vectors 𝐢 hat and 𝐣 hat, we can see that since 𝐢 hat defines the 𝑥-direction, then 𝑎 subscript 𝑥 must be the 𝑥-component of 𝑎. And likewise, since 𝐣 hat defines the 𝑦-direction, then 𝑎 subscript 𝑦 must be the 𝑦-component of 𝑎. In order to write the vector 𝐀 in component form, what we need to do is to find the values of its 𝑥- and 𝑦-components, 𝑎 subscript 𝑥 and 𝑎 subscript 𝑦.

Let’s begin with the 𝑥-component, so that’s 𝑎 subscript 𝑥. To find the value of 𝑎 subscript 𝑥, we need to trace a line vertically from the tip of vector 𝐀 until we get to the 𝑥-axis. Then, we need to count the number of squares from the origin to the point that we’ve found this line meets the axis. Counting these squares, we find that there are one, two, three, four of them.

However, the positive 𝑥-direction as defined by the unit vector 𝐢 hat is to the right. And we’ve counted our squares in the direction to the left. That means that we’ve counted our four squares in the negative 𝑥-direction. And so the 𝑥-component of 𝐀 is equal to negative four. This 𝑥-component of 𝐀 that we found is the value of 𝑎 subscript 𝑥. So we found then that 𝑎 subscript 𝑥 is equal to negative four.

Now, we need to do the same thing for the vector’s 𝑦-component. That’s 𝑎 subscript 𝑦. For this, we need to trace across from the tip of the vector until we get to the 𝑦-axis. Then, counting the squares between the origin and the point where this line meets the axis, we find that there are one, two, three, four, five, six, seven, eight squares.

But the positive 𝑦-direction, as defined by this unit vector 𝐣 hat, is upward. And we’ve counted our squares in the opposite direction to this, downward. So we’ve counted the squares in the negative 𝑦-direction, which means that the 𝑦-component of the vector is equal to negative eight. Since this 𝑦-component is our value of 𝑎 subscript 𝑦, then we have that 𝑎 subscript 𝑦 is equal to negative eight.

Now that we found the values for both 𝑎 subscript 𝑥 and 𝑎 subscript 𝑦, all that’s left to do is to substitute these values into the expression for the vector 𝐀. When we do this, we find that 𝐀 is equal to negative four, that’s our value for 𝑎 subscript 𝑥, multiplied by 𝐢 hat plus negative eight, our value of 𝑎 subscript 𝑦, multiplied by 𝐣 hat. We can write this expression more simply as negative four 𝐢 hat minus eight 𝐣 hat. This is our final expression for the vector 𝐀 in component form.