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
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.