Video: Finding the Magnitude of the Vector Joining the End Points of Two Given Vectors

Given that 𝐀𝐁 = −5𝐢 + 2𝐣 − 4𝐤 and 𝐁𝐂 = 4𝐢 + 4𝐣 + 6𝐤, determine |𝐀𝐂|.

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Video Transcript

Given that vector 𝐀𝐁 is equal to negative five 𝐢 plus two 𝐣 minus four 𝐤 and vector 𝐁𝐂 is equal to four 𝐢 plus four 𝐣 plus six 𝐤, determine the magnitude of vector 𝐀𝐂.

In this type of question, it is worth drawing a diagram first. This will hopefully ensure that our direction and signs are correct. We are given three points 𝐴, 𝐵, and 𝐶. We can join these to form a triangle. In the question, we are given the value of vector 𝐀𝐁. We are also given the value of vector 𝐁𝐂. Our aim is to calculate the magnitude of vector 𝐀𝐂. Therefore, our first step is to work out vector 𝐀𝐂.

We can see that one way to get from point 𝐴 to point 𝐶 is via point 𝐵. Therefore, vector 𝐀𝐂 is equal to vector 𝐀𝐁 plus vector 𝐁𝐂. Vector 𝐀𝐂 is, therefore, equal to negative five 𝐢 plus two 𝐣 minus four 𝐤 plus four 𝐢 plus four 𝐣 plus six 𝐤.

We can add two vectors by adding the individual components separately. Negative five 𝐢 plus four 𝐢 is equal to negative 𝐢. Two 𝐣 plus four 𝐣 is equal to six 𝐣. Finally, negative four 𝐤 plus six 𝐤 is equal to two 𝐤. Vector 𝐀𝐂 is equal to negative 𝐢 plus six 𝐣 plus two 𝐤.

The magnitude of any vector can be found by squaring the 𝐢-, 𝐣-, and 𝐤-components, finding their sum, and then square rooting the answer. This means that the magnitude of vector 𝐀𝐂 is the square root of negative one squared plus six squared plus two squared. Negative one squared is equal to one, six squared is 36, and two squared is equal to four. One, 36, and four sum to 41. Therefore, the magnitude of vector 𝐀𝐂 is the square root of 41.

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