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Question Video: Using the Norm of Vectors to Solve a Problem Mathematics

Given that 𝐀 + 𝐁 = 〈−2, 4, 3〉 and 𝐀 = 〈3, 5, 3〉, determine |𝐁|.

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Video Transcript

Given that vector 𝐀 plus vector 𝐁 is equal to negative two, four, three and vector 𝐀 is equal to three, five, three, determine the magnitude of vector 𝐁.

We recall that when adding two vectors, we simply add the 𝐢-, 𝐣-, and 𝐤-components separately. The vector 𝐀 plus 𝐁 is equal to the vector 𝐀 plus the vector 𝐁. If we let vector 𝐁 have 𝐢-, 𝐣-, and 𝐤-components 𝑥, 𝑦, and 𝑧, respectively, then negative two, four, three is equal to three, five, three plus 𝑥, 𝑦, 𝑧. We can then subtract vector 𝐀 from both sides of this equation. The left-hand side becomes negative two, four, three minus three, five, three.

Negative two minus three is equal to negative five. Therefore, our 𝐢-component of vector 𝐁 is negative five. Four minus five is equal to negative one, so the 𝐣-component is negative one. Finally, three minus three is equal to zero. Vector 𝐁 is, therefore, equal to negative five, negative one, zero.

We can calculate the magnitude of this vector by squaring each of the components, finding their sum, and then square rooting. The magnitude of vector 𝐁 is equal to the square root of negative five squared plus negative one squared plus zero squared. Negative five squared is 25, negative one squared is one, and zero squared is equal to zero. The magnitude of vector 𝐁 is, therefore, equal to the square root of 26.

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