Video Transcript
𝐕 and 𝐖 are two vectors, where
vector 𝐕 is equal to negative one, five, negative two and vector 𝐖 is equal to
three, one, one. Comparing the magnitude of vector
𝐕 minus vector 𝐖 and the magnitude of vector 𝐕 minus the magnitude of vector 𝐖,
which quantity is larger?
We recall that the magnitude of any
vector 𝐀 with components 𝑥, 𝑦, and 𝑧 is equal to the square root of 𝑥 squared
plus 𝑦 squared plus 𝑧 squared. We find the sum of the squares of
the individual components and then square root the answer. This means that the magnitude of
vector 𝐕 is equal to the square root of negative one squared plus five squared plus
negative two squared. This is equal to the square root of
one plus 25 plus four. The magnitude of vector 𝐕 is equal
to the square root of 30. We can repeat this process for
vector 𝐖. This is equal to the square root of
three squared plus one squared plus one squared, which simplifies to the square root
of nine plus one plus one, which is the square root of 11. The magnitude of vector 𝐖 is the
square root of 11.
Before calculating a numerical
value for the magnitude of vector 𝐕 minus the magnitude of vector 𝐖, we will
firstly work out vector 𝐕 minus vector 𝐖. We know that to subtract two
vectors, we subtract the corresponding components. Negative one minus three is equal
to negative four. Five minus one is equal to positive
four. And negative two minus one is equal
to negative three. Vector 𝐕 minus vector 𝐖 is equal
to negative four, four, negative three. The magnitude of this is equal to
the square root of negative four squared plus four squared plus negative three
squared. This simplifies to the square root
of 16 plus 16 plus nine. The magnitude of vector 𝐕 minus
vector 𝐖 is therefore equal to the square root of 41.
We now need to calculate the
decimal values of the magnitude of vector 𝐕 minus vector 𝐖 and the magnitude of
vector 𝐕 minus the magnitude of vector 𝐖. The square root of 41 is equal to
6.4031 and so on. The square root of 30 minus the
square root of 11 is equal to 2.1606 and so on. This means that the square root of
41 is greater than the square root of 30 minus the square root of 11. We can therefore conclude that the
magnitude of vector 𝐕 minus vector 𝐖 is greater than the magnitude of vector 𝐕
minus the magnitude of vector 𝐖. The larger quantity is the
magnitude of vector 𝐕 minus vector 𝐖.
