Video Transcript
𝐕 and 𝐖 are two
vectors, where 𝐕 is the vector negative one, five, negative two and 𝐖 is the
vector three, one, one. Comparing the magnitude of 𝐕 plus
𝐖 and the magnitude of vector 𝐕 plus the magnitude of vector 𝐖, which quantity is
larger?
In this question, we’re given two
three-dimensional vectors, the vector 𝐕 and the vector 𝐖. And we’re told the components of
these two vectors. We need to determine which of two
quantities is larger, the magnitude of vector 𝐕 plus vector 𝐖 or the magnitude of
vector 𝐕 plus the magnitude of vector 𝐖. There are many different ways of
answering this question. Since we’re given the components of
vector 𝐕 and vector 𝐖, let’s start by calculating both of these
expressions.
Let’s start with the magnitude of
the sum of these two vectors. That’s the magnitude of the vector
negative one, five, negative two plus the vector three, one, one. To evaluate this expression, we’re
going to need to add these two vectors together. To do this, we just need to add the
corresponding components together. This gives us the magnitude of the
vector negative one plus three, five plus one, negative two plus one. And if we evaluate the expressions
for each of these components, we get the magnitude of the vector two, six, negative
one.
To evaluate the magnitude of this
vector, we recall the magnitude of a vector is the square root of the sum of the
squares of its components. So the magnitude of the vector 𝑎,
𝑏, 𝑐 will be equal to the square root of 𝑎 squared plus 𝑏 squared plus 𝑐
squared. Applying this to the vector two,
six, negative one, we get that its magnitude is equal to the square root of two
squared plus six squared plus negative one all squared, which if we simplify the
expression inside of our square root symbol, we get the square root of 41.
Next, we want to compare this to the
magnitude of vector 𝐕 plus the magnitude of vector 𝐖. That’s the magnitude of the vector
negative one, five, negative two added to the magnitude of the vector three, one,
one. Evaluating the magnitude of each of
these vectors, we get the square root of negative one squared plus five squared plus
negative two all squared plus the square root of three squared plus one squared plus
one squared, which we can simplify to give us the square root of 30 plus the square
root of 11. Finally, the question wants us to
determine which of these two quantities is larger. We’ll do this by evaluating both of
these expressions to one decimal place. We get 6.4 and 8.8,
respectively. Therefore, since 8.8 is larger than
6.4, we’ve shown the magnitude of 𝐕 plus the magnitude of 𝐖 is larger than the
magnitude of 𝐕 plus 𝐖.
Now we could stop here. However, there is another way of
showing this is true in general. Remember, we can represent any
vector graphically and we can also add two vectors together graphically. For example, suppose we have the
vector 𝐕 and vector 𝐖 as shown. In this graphical interpretation,
the vectors represent displacement, so we can represent the vector 𝐕 plus 𝐖 as
shown. And of course, when we represent
these vectors graphically, their magnitude is their length. Using this diagram, we can find a
link between the magnitude of vector 𝐕 plus 𝐖 and the magnitude of vector 𝐕 plus
the magnitude of vector 𝐖. The magnitude of vector 𝐕 plus 𝐖
is going to be the length of the line segment from the initial point of this vector
to its terminal point. It’s just the length of the base of
this triangle.
However, the magnitude of vector 𝐕
added to the magnitude of vector 𝐖 is going to be the length of the other two sides
of our triangle added together. And we know the shortest distance
between two points is the straight line between them. This gives us a graphical intuition
of the following inequality. The magnitude of vector 𝐕 plus
vector 𝐖 will be less than or equal to the magnitude of vector 𝐕 plus the
magnitude of vector 𝐖. This is often referred to as the
triangle inequality. And it’s true in any number of
dimensions and we can prove this algebraically. Finally, we could use this to
answer our question by using the fact that our inequality will be strict if neither
of our two vectors 𝐕 or 𝐖 is the zero vector.
We could therefore use this
inequality to conclude that if 𝐕 is the vector negative one, five, negative two and
𝐖 is the vector three, one, one, then the magnitude of vector 𝐕 plus the magnitude
of vector 𝐖 will be bigger than the magnitude of vector 𝐕 plus vector 𝐖.
