Video Transcript
If 𝐀 equals three 𝐢 plus four 𝐣,
𝐁 equals four 𝐣, and 𝐂 equals six 𝜋 over 10, then the magnitude of 𝐀 plus the
magnitude of 𝐁 plus the magnitude of 𝐂 equals blank. (A) 15, (B) six, (C) 11, (D)
10.
Okay, so here we have these three
vectors 𝐀, 𝐁, and 𝐂. And we want to find the sum of
their magnitudes. In general, for a vector given in
terms of its 𝑥- and 𝑦-components, its magnitude is equal to the square root of the
sum of the squares of those components. We can apply this rule to vectors
𝐀 and 𝐁 to solve for their respective magnitudes. The magnitude of 𝐀 is equal to the
square root of three squared plus four squared. That’s equal to the square root of
25 or simply five. And then we could apply the same
rule to vector 𝐁. But notice that since this just has
one component, its magnitude is equal to the magnitude of that component. Vector 𝐁 has a magnitude of
four.
Then lastly, we want to calculate
the magnitude of vector 𝐂, which we see is not given in its Cartesian components,
but rather in polar form. When a vector is given in this
form, we’re being told the radial distance of the vector from some origin, in other
words, the vector’s length, along with the direction that the vector points. So the nice thing is that for a
vector given in this form, we already know its magnitude. It’s the radial distance 𝑟. Looking at vector 𝐂 then, we can
simply read off its magnitude. It has a magnitude of six. And so when we add these three
magnitudes together, we get five plus four plus six, which is 15. Given these three vectors 𝐀, 𝐁,
and 𝐂, the sum of their magnitudes is 15.
