Video Transcript
Consider the vectors ๐ฎ equals two,
three and ๐ฏ equals four, six. What is the magnitude of vector
๐ฎ? What is the magnitude of vector
๐ฏ? What is the magnitude of vector ๐ฎ
plus vector ๐ฏ? In all three questions we need to
give our answer to two decimal places where appropriate.
We recall that the magnitude of any
vector ๐ฐ can be found by square rooting ๐ squared plus ๐ squared, where ๐ and ๐
are the two components of the vector. In vector ๐ฎ, ๐ is equal to two
and ๐ is equal to three. Whereas in vector ๐ฏ, ๐ is equal
to four and ๐ is equal to six. The magnitude of vector ๐ฎ 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 ๐ฎ is equal to the square
root of 13. We would often leave this in surd
or radical form. However, in this case, weโre 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 ๐ฎ to two
decimal places is 3.61. We can repeat this process to
calculate the magnitude of vector ๐ฏ. Four squared is equal to 16, and
six squared is equal to 36. Therefore, the magnitude of vector
๐ฏ 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, weโll
round down. The magnitude of vector ๐ฏ is
therefore equal to 7.21.
The final part of our question asks
us to work out the magnitude of ๐ฎ plus ๐ฏ. Our first step here will be to
calculate the vector ๐ฎ plus ๐ฏ. 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 ๐ฎ plus ๐ฏ 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 ๐ฎ plus
๐ฏ 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 ๐ฎ plus ๐ฏ is
equal to 10.82.
When looking at our three answers,
you might think youโve spotted a pattern, as 3.61 plus 7.21 is equal to 10.82. This suggests that the magnitude of
๐ฎ plus ๐ฏ is equal to the magnitude of ๐ฎ plus the magnitude of ๐ฏ. This, however, is not normally the
case. The only reason this works in this
question is that vector ๐ฏ is actually a multiple of vector ๐ฎ. Two multiplied by two is equal to
four. And three multiplied by two is
equal to six.
Therefore, vector ๐ฏ is actually
two lots or two multiplied by vector ๐ฎ. This in turn means that the
magnitude of vector ๐ฏ is twice the magnitude of vector ๐ฎ. Root 52 is equal to two root
13. The magnitude of ๐ฎ plus ๐ฏ in this
question becomes three times the magnitude of vector ๐ฎ. It is important to note, however,
as previously mentioned, this will not hold for the majority of vector
questions.
