Video Transcript
True or false. Given that vector 𝐀 is equal to one, two, three; vector 𝐁 is equal to two, six, two; and vector 𝐂 is equal to zero, two, zero, then the magnitude of vector 𝐀 plus vector 𝐁 plus vector 𝐂 is greater than the magnitude of vector 𝐀 minus vector 𝐁 minus vector 𝐂.
In this question, we begin by adding and subtracting vectors. In order to do this, we need to add or subtract their corresponding components. On the left-hand side, we need to add vector 𝐀, vector 𝐁, and vector 𝐂. Adding the 𝑥-components gives us three, adding the 𝑦-components gives us 10, and adding the 𝑧 or 𝑧-components gives us five. Vector 𝐀 plus vector 𝐁 plus vector 𝐂 is equal to three, 10, five.
Our next step is to calculate the magnitude of this vector. We know that we can calculate the magnitude of a vector by finding the square root of the sum of the squares of its individual components. The magnitude of vector 𝐀 plus vector 𝐁 plus vector 𝐂 is equal to the square root of three squared plus 10 squared plus five squared. Three squared is equal to nine, 10 squared is 100, and five squared is equal to 25. As nine plus 100 plus 25 is equal to 134, the magnitude of vector 𝐀 plus vector 𝐁 plus vector 𝐂 is the square root of 134.
We will now repeat this process for the right-hand side. We begin by calculating vector 𝐀 minus vector 𝐁 minus vector 𝐂. One minus two minus zero is equal to negative one. Two minus six minus two is equal to negative six. Finally, three minus two minus zero is equal to one. Once again, we need to find the magnitude of this vector. This is equal to the square root of negative one squared plus negative six squared plus one squared. Negative one squared and one squared are both equal to one. Negative six squared is equal to 36. As one, 36, and one sum to 38, the magnitude of vector 𝐀 minus vector 𝐁 minus vector 𝐂 is equal to the square root of 38.
The inequality sign in the question told us that the left-hand side was greater than the right-hand side. The square root of 134 is indeed greater than the square root of 38. We can, therefore, conclude that given the vectors 𝐀, 𝐁, and 𝐂, then the magnitude of vector 𝐀 plus vector 𝐁 plus vector 𝐂 is greater than the magnitude of vector 𝐀 minus vector 𝐁 minus vector 𝐂. The correct answer is true.
