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.