High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.A.1
1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
Sometimes, a number is enough. You know, that dream you have, where all you seem to hear is "Thirty million dollars!" The rest of the dream is a blur, but that one single number is enough to evoke images of private jets, vacations in the Caribbean, and that long-awaited Netflix subscription.
Other times, numbers need a little help. They need direction. That's where vectors come in.
Students should know that vectors are directed line segments, having both magnitude and direction. That means a vector not only tells you how fast the wind is blowing, but also the direction it's blowing in.
They should also know the symbols for vectors. (Mathematicians love symbols and shortcuts, and vectors are no exception.) Instead of calling a vector AB, which is far too much work, mathematicians prefer to call it v (for vector, obviously). In general, boldfaced lowercase letters represent vectors.
If students want to refer to the magnitude, or length, of the vector, they put it inside a double absolute value: ||v||.
Two vectors that have the same magnitude and direction are said to be equivalent (though we use an equal sign to represent equivalency). Equivalent vectors can be thought of as congruent segments on parallel lines: same direction and same length.
The direction of a vector is determined by finding the slope of the segment between its initial point and its terminal point, while the magnitude of a vector is the same as the distance between its endpoints. A vector is said to be in standard form if its initial point is the origin.