Question Video: Expressing a Given Vector in Terms of Fundamental Unit Vectors Mathematics

Given that 𝐴 = (0, 0, 5), express vector 𝚨𝐎 in terms of the unit vectors 𝐒, 𝐣, and 𝐀.

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Video Transcript

Given that point 𝐴 has coordinates zero, zero, five, express vector 𝚨𝐎 in terms of the unit vectors 𝐒, 𝐣, and 𝐀.

Let’s begin by considering the three-dimensional coordinate plane as shown. We are told that point 𝐴 has coordinates zero, zero, five. This means that its π‘₯-coordinate is zero, its 𝑦-coordinate is zero, and its 𝑧-coordinate is five. The point, therefore, lies on the 𝑧-axis. The unit vectors 𝐒 hat, 𝐣 hat, and 𝐀 hat are the vectors of magnitude one in the positive π‘₯-, 𝑦-, and 𝑧-directions.

In this question, we need to express the vector from point 𝐴 to point 𝑂, where 𝑂 is the origin with coordinate zero, zero, zero. To travel from point 𝐴 to point 𝑂, we travel zero units in the π‘₯-direction, zero units in the 𝑦-direction, and negative five units in the 𝑧-direction. The vector 𝚨𝐎 is therefore equal to zero 𝐒 plus zero 𝐣 plus negative five 𝐀. This simplifies to negative five 𝐀.

An alternative method here would be to find the vector 𝐎𝚨 first. Any vector starting at the origin will have components equal to the coordinates of the endpoint. As point 𝐴 has coordinates zero, zero, five, the vector 𝐎𝚨 will be equal to zero 𝐒 plus zero 𝐣 plus five 𝐀. This simplifies to five 𝐀. We know that reversing the direction of a vector does not alter its magnitude. Vector 𝚨𝐎 is equal to the negative of vector 𝐎𝚨. This confirms, as we have already shown, that vector 𝚨𝐎 is equal to negative five 𝐀.

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