Video Transcript
Consider the vectors 𝐮 equals two,
three and 𝐯 equals four, six. What is the magnitude of vector
𝐮? What is the magnitude of vector
𝐯? What is the magnitude of vector 𝐮
plus vector 𝐯? In all three questions we need to
give our answer to two decimal places where appropriate.
We recall that the magnitude of any
vector 𝐰 can be found by square rooting 𝑎 squared plus 𝑏 squared, where 𝑎 and 𝑏
are the two components of the vector. In vector 𝐮, 𝑎 is equal to two
and 𝑏 is equal to three. Whereas in vector 𝐯, 𝑎 is equal
to four and 𝑏 is equal to six. The magnitude of vector 𝐮 is
therefore equal to the square root of two squared plus three squared. As two squared is equal to four and
three squared is equal to nine, the magnitude of vector 𝐮 is equal to the square
root of 13. We would often leave this in surd
or radical form. However, in this case, we’re asked
to give our answer to two decimal places. The square root of 13 is equal to
3.605551 and so on.
To round to two decimal places, our
key or deciding number will be the first five. This will round our answer up. The magnitude of vector 𝐮 to two
decimal places is 3.61. We can repeat this process to
calculate the magnitude of vector 𝐯. Four squared is equal to 16, and
six squared is equal to 36. Therefore, the magnitude of vector
𝐯 is the square root of 52. Typing this into the calculator
gives us 7.211102 and so on. This time our deciding number is a
one. As this is less than five, we’ll
round down. The magnitude of vector 𝐯 is
therefore equal to 7.21.
The final part of our question asks
us to work out the magnitude of 𝐮 plus 𝐯. Our first step here will be to
calculate the vector 𝐮 plus 𝐯. We do this by adding the
corresponding components. Two plus four is equal to six, and
three plus six is equal to nine. We can then calculate the magnitude
of 𝐮 plus 𝐯 in the same way. This is equal to the square root of
six squared plus nine squared. Six squared is equal to 36, and
nine squared is equal to 81. Therefore, the magnitude of 𝐮 plus
𝐯 is equal to the square root of 117. Typing this into the calculator
gives us 10.816653. The deciding number here is a six,
and anything five or greater means that we round up. The magnitude of 𝐮 plus 𝐯 is
equal to 10.82.
When looking at our three answers,
you might think you’ve spotted a pattern, as 3.61 plus 7.21 is equal to 10.82. This suggests that the magnitude of
𝐮 plus 𝐯 is equal to the magnitude of 𝐮 plus the magnitude of 𝐯. This, however, is not normally the
case. The only reason this works in this
question is that vector 𝐯 is actually a multiple of vector 𝐮. Two multiplied by two is equal to
four. And three multiplied by two is
equal to six.
Therefore, vector 𝐯 is actually
two lots or two multiplied by vector 𝐮. This in turn means that the
magnitude of vector 𝐯 is twice the magnitude of vector 𝐮. Root 52 is equal to two root
13. The magnitude of 𝐮 plus 𝐯 in this
question becomes three times the magnitude of vector 𝐮. It is important to note, however,
as previously mentioned, this will not hold for the majority of vector
questions.
