Video Transcript
If vector 𝐀 is equal to nine 𝐢
minus seven 𝐣 plus eight 𝐤 and vector 𝐁 is equal to eight 𝐢 minus three 𝐣 plus
four 𝐤, find vector 𝐀 minus two-thirds of vector 𝐁.
In this question, we will need to
recall how to subtract vectors and also how to multiply a vector by a scalar. Let’s begin by calculating
two-thirds of vector 𝐁.
In order to multiply any vector by
a scalar, we simply multiply each of the components by that scalar. Two-thirds multiplied by eight is
equal to sixteen-thirds. Therefore, two-thirds multiplied by
eight 𝐢 is sixteen-thirds 𝐢. Multiplying two-thirds by negative
three 𝐣 gives us negative two 𝐣. Finally, multiplying two-thirds by
four 𝐤 gives us eight-thirds 𝐤. Multiplying vector 𝐁 by the scalar
two-thirds gives us sixteen-thirds 𝐢 minus two 𝐣 plus eight-thirds 𝐤.
We need to subtract this vector
from vector 𝐀. In order to subtract two vectors,
we simply subtract the corresponding components. Nine 𝐢 minus sixteen-thirds 𝐢 is
equal to eleven-thirds 𝐢. This is because nine can be
rewritten as twenty-seven thirds or 27 over three. Subtracting the 𝐣-components gives
us negative five 𝐣, as negative seven 𝐣 minus negative two 𝐣 is the same as
negative seven 𝐣 plus two 𝐣.
Finally, eight 𝐤 minus
eight-thirds 𝐤 is equal to sixteen-thirds 𝐤. We know this as we can rewrite
eight as twenty-four thirds. And twenty-four thirds minus
eight-thirds is equal to sixteen-thirds. Vector 𝐀 minus two-thirds of
vector 𝐁 is therefore equal to eleven-thirds 𝐢 minus five 𝐣 plus sixteen-thirds
𝐤.
