Video: Performing Operations on Vectors Algebraically in 2D

If vector 𝐀𝐁 = 7𝑖 + 6𝑗 and vector 𝐁𝐂 = 𝑖, then |𝐀𝐂| = ＿.

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Video Transcript

If vector 𝐀𝐁 is equal to seven 𝑖 plus six 𝑗 and vector 𝐁𝐂 is equal to 𝑖, then the magnitude of vector 𝐀𝐂 is equal to blank.

We recall that the vector seven 𝑖 plus six 𝑗 can be written using triangular brackets as seven, six. The vector 𝑖 can be written one, zero, as there is no vertical component 𝑗. We also recall that when dealing with vectors, the vector 𝐀𝐂 is equal to the vector 𝐀𝐁 plus the vector 𝐁𝐂. In this question, we can calculate the vector 𝐀𝐂 by adding the vector six, one and the vector one, zero.

When adding two vectors, we add the horizontal and vertical components separately. Seven plus one is equal to eight. Six plus zero is equal to six. Therefore, vector 𝐀𝐂 is equal to eight, six. This is not the final answer to this question though, as we need to calculate the magnitude or modulus of vector 𝐀𝐂. In order to do this, we need to know the following rule. If vector 𝐮 is equal to 𝑥, 𝑦, then the magnitude of vector 𝐮 is equal to the square root of 𝑥 squared plus 𝑦 squared. We use the Pythagorean theorem to calculate the magnitude or size of vector 𝐮 by squaring the horizontal and vertical components, finding their sum, and then square rooting the answer.

The magnitude of vector 𝐀𝐂 is equal to the square root of eight squared plus six squared. Eight squared is equal to 64, and six squared is 36. Adding these gives us the square root of 100, which is equal to 10. If vector 𝐀𝐁 is equal to seven 𝑖 plus six 𝑗 and vector 𝐁𝐂 is equal to 𝑖, then the magnitude of vector 𝐀𝐂 is 10.