Question Video: Finding the Horizontal Component of a Vector Shown on a Grid Physics

The vector 𝚨 can be written in the form 𝑎_(𝑥)𝐢 + 𝑎_(𝑦)𝐣. What is the value of 𝑎_(𝑥), What is the value of 𝑎_(𝑦)?

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Video Transcript

The vector 𝚨 can be written in the form 𝑎 sub 𝑥 𝐢 hat plus 𝑎 sub 𝑦 𝐣 hat. What is the value of 𝑎 sub 𝑥, and what is the value of 𝑎 sub 𝑦?

Let’s first recall the meanings of 𝐢 hat and 𝐣 hat. These are unit vectors. And since our vector is drawn onto a grid, we can define one grid square to be one unit such that one unit in the horizontal direction is equal to 𝐢 hat and one unit in the vertical direction is equal to 𝐣 hat. So let’s first look at what is the value of 𝑎 sub 𝑥. 𝑎 sub 𝑥 is the number that we need to multiply by the unit vector 𝐢 hat to make the horizontal component of the vector 𝚨.

So we start from the tail of vector 𝚨, and note that it touches the vertical axis. So we start from the vertical axis and count horizontally towards the tip of vector 𝚨 one, two units. But note that we went to the left from the axis, whereas the unit vector 𝐢 hat increases to the right, which means that the number of units in the horizontal direction is negative two. So the answer to “What is the value of 𝑎 sub 𝑥?” is negative two.

Next, let’s look at the value of 𝑎 sub 𝑦. 𝑎 sub 𝑦 is the number we need to multiply by the unit vector in the vertical direction 𝐣 hat to get the vertical component of the vector 𝚨. So again, we start from the tail of vector 𝚨, and note that it touches the horizontal axis. So we start from the horizontal axis and count vertically towards the tip of vector 𝚨 one, two, three, four, five units. And note that we counted upwards from the axis, which is the same direction as the unit vector 𝐣 hat. So the number of units in the vertical direction is positive five, which makes the answer to “What is the value of 𝑎 sub 𝑦?” five.