Whenever you get a new mathematical object, you need to learn what you can do with it.
So ... what operations can be performed with vectors?
Let $\,k\,$ be a real number (a scalar).
Then, $\,k\vec v\,$ is a vector.
That is, a scalar times a vector produces a vector.
In other words, a real number times a vector produces a vector.
The image below illustrates the relationship between an original vector $\,\vec v\,$ and various scaled versions:
Multiplying a vector by a positive number $\,k\,$:
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Multiplying a vector by a negative number $\,k\,$:
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Multiplying any vector by the real number $\,0\,$ produces the zero vector. For all vectors $\,\vec v\,$, $$\cssId{s27}{-\vec v\, = -1\cdot\vec v}$$ is the opposite of $\,\vec v\,$. |
To find the size of a scaled vector, you multiply together two numbers:
Keep in mind:
$\|3\vec v\|$ | $=$ | $|3|\cdot\|\vec v\|$ | $=$ | $3\|\vec v\|$ |
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$\|-3\vec v\|$ | $=$ | $|-3|\cdot\|\vec v\|$ | $=$ | $3\|\vec v\|$ |
Adding the arrow representations of vectors is done using the ‘head-of-first to tail-of-second’ rule.
This is usually abbreviated as ‘head-to-tail addition’.
Here's how to add $\,\vec u\,$ to $\,\vec v\,$:
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Vector Addition is Commutative:
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Vector Addition is Associative:
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To subtract a vector, just add its opposite: $$\,\cssId{s86}{\vec u - \vec v := \vec u + (-\vec v)}\,$$ (Remember that ‘$\,:=\,$’ means ‘equals, by definition’).
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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