Often used to represent velocity, distance traveled, or force, vectors are line segments defined by two properties: length and direction.
Graphically, a vector is shown as a line segment with an arrow on one end, to show its direction:
In the preceding figure, vector u, which can also be written , has initial point (–2, –2), and terminal point (6, 4). When you see a vector equated to a coordinate pair, say u = (8, 6), like in the figure above, the first coordinate is called the x-component of the vector and the second coordinate is the y-component.
The length of a vector, also known as its magnitude, equals the square root of the sum of the squares of its components. Basically, it’s a restatement of the Pythagorean theorem. The length of the vector above, for example, is = 10.
The length of a vector can be found if you know its initial point and terminal point, or if you know the components of the vector.
Adding and Subtracting Vectors
To add or subtract vectors, just add or subtract their respective components. For example, if a = (2, 7) and b = (–5, 2), a + b = (2 + –5, 7 + 2) = (–3, 9). In order to graph this, you first place the second vector so that its tail starts at the tip of the first vector. Then close the triangle by drawing a vector from the tail of the first vector to the tip of the second vector. It looks like this:
Subtracting vectors is the same process, except that you reverse the direction of the subtracted vector: ab = a + (–b) = (2 – (–5), 7 – 2) = (7, 5).
Multiplying by a Scalar
Occasionally, the Math IIC will ask you to multiply a vector by a scalar, which is a number that has magnitude but no direction. To answer this type of question, just multiply each component of the vector by the scalar. If u = (3, 4), cu = (3c, 4c).
Try an example problem:
If u = (1, –5) and v = (4, 2), what is 3u – 2v?
It’s easy—just multiply and then subtract: 3u – 2v = 3(1, –5) – 2(4, 2) = (3, –15) – (8, 4) = (–5, –19).
Calculating the length, sum, difference, or product of vectors and scalars is the extent of what the Math IIC will ask you on this subject.
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