Video Transcript
If vector 𝐀 is equal to five,
negative three and vector 𝐁 is equal to two, one, then the magnitude of 𝐀 plus
three 𝐁 is equal to what length units.
In this question, we are given the
two-dimensional vectors 𝐀 and 𝐁 in terms of their 𝑥- and 𝑦-components. We are asked to calculate the
magnitude of vector 𝐀 plus three multiplied by vector 𝐁. We will do this in three steps,
firstly by using scalar multiplication to calculate three 𝐁. When multiplying any vector by a
scalar, we simply multiply each of the individual components by that scalar. This means that three multiplied by
the vector two, one gives us the vector six, three. Our next step is to add this to
vector 𝐀. And we will do this using the
process of vector addition. We do this by adding the
corresponding components separately, giving us the vector 11, zero.
Our final step is to find the
magnitude of this vector. As the 𝑦-component of our vector
is zero, there is a shortcut here. However, we will begin by looking
at how we calculate the magnitude of any two-dimensional vector. If vector 𝐮 is equal to 𝑥, 𝑦,
then the magnitude of vector 𝐮 is equal to the square root of 𝑥 squared plus 𝑦
squared. We find the sum of the squares of
the individual components and then square root our answer. This means that the magnitude of
the vector 11, zero is equal to the square root of 11 squared plus zero squared. This is equal to the square root of
121, which is equal to 11. If vector 𝐀 is equal to five,
negative three, vector 𝐁 is equal to two, one, then the magnitude of 𝐀 plus three
𝐁 is equal to 11 length units.
As previously mentioned, there is a
shortcut to calculate the magnitude when one of the components equals zero. In this question, the 𝑦-component
was equal to zero. This means that the vector 11, zero
moves 11 units in the positive 𝑥-direction. As the magnitude of a vector is its
length, this confirms that the magnitude of the vector 11, zero is 11 length
units. When a two-dimensional vector has
one of its components equal to zero, then the magnitude of that vector will be equal
to the nonzero component.
