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This essential trains for: Math Kangaroo 9-10, AMC-10, AMC-12, AIME.
Vectors are mathematical objects that represent three types of information: a number (magnitude, length, absolute value), a support line, a specific orientation of the support line. These last two notions are often lumped up as "direction."
Numbers (objects that have only size) are also called scalars.
A vector has a simple graphical representation as an arrow either in the plane (2D) or in space (3D). The point where the arrow starts is called initial point. The point where the arrow ends is called terminal point.
Vectors that have the same magnitude and parallel directions are called equipolent.
If a vector is considered to be the same as any other vectors equipolent to it, it is called a free vector. Otherwise, if its initial and terminal points are fixed, it is called a bound vector.
We will denote a vector with a symbol that is topped by an arrow:
or with a bolded symbol.
The magnitude of the vector is simply a number (scalar) and will be denoted by a symbol without arrow.
Operations with Vectors
Vector Addition applied to two vectors produces a third vector called resultant.
The operation is defined as follows:
Using the cosine law, the magnitude of the resultant vector is:
where the angle supposes that the vectors to be added share the initial point.
Subtracting vectors is the same as addition with an opposite vector:
By placing a vector in a Cartesian system of coordinates, we can use the definition of vector addition to define a vector's components, i.e. its vector projections on the axes of coordinates.
If the vector is positioned such as to make an angle θ with the positive direction of the x-axis, the magnitudes of the components can be calculated using trigonometric ratios:
Of course, the magnitude of the vector a and the magnitudes of the its components satisfy the Pythagorean theorem:
or, in 3D:
Multiplication by a scalar affects only the magnitude of a vector:
We can separate the information about the magnitude from the information about direction by using a unit vector. A unit vector is a vector of magnitude 1 that encapsulates the information about some direction, i.e. it tells us in which direction to go but not how far to go in that direction (unless we want to go a single unit away):
In the figure, the unit vector u encapsulates the direction of the support line. The vector v can be obtained by scaling the unit vector u by a positive scalar factor, while the vector w can be obtained by scaling the unit vector u by a negative scalar factor.
Unit vectors for the axes of a Cartesian system of coordinates have special names:
Vector addition using components
Addition/subtraction of vectors can now be performed using components.
Write the vectors a and b using components. Note that the components are also vectors and that the 'plus' operator represents vector addition, not addition of scalars (numbers):
It is easy to add components that lie in the same direction, since they make an angle of zero; therefore, the magnitude of such a sum is just the regular addition of the magnitudes:
We can represent the components using the unit vectors for the respective axes:
Or, if the vectors a and b are represented in 3D:
Note that the "multiplication" operation:
is the multiplication of a vector by a scalar, whereas the "addition" operation:
is vector addition.
It is important to realize that these operators are defined as above and are not the same as the operators that are symbolized in the same manner but operate on numbers.
The Dot Product of Vectors is an operation that takes two vectors as operands and results in a scalar (number):
If we place the two vectors in a Cartesian system of coordinates, we can write the definition using components and unit vectors:
Using the distributive property and the associative property:
We know the angles between the unit vectors of the axes, hence:
Or, in 3D:
The dot product must give the same result regardless of how we represent the vectors:
Where from we can find a useful formula for the angle between two vectors (segments):
The above formula in 3D:
where we do not forget that the absolute values of the vectors must be used:
The Cross Product of Vectors is an operation that takes two vectors as operands and results in a vector. The resulting vector is perpendicular on both operands, i.e. it lies on the support line:
The direction on the support line in which the cross product points depends on the order of the operands. The cross product operator is not commutative.
To find its direction, use the right hand rule: use your right hand fingers to rotate the first operand towards the second one; the thumb will point in the direction of the cross-product:
The magnitude of the cross-product is defined as:
Note that the magnitude is equal to double the area of the triangle formed by the two vectors:
By writing the cross product using coordinates, we find a useful formula for the area of a triangle in 2D or in 3D space:
Using the distributive and the associative property:
Since we know the angles between the unit vectors, we find:
where we have used the unit vector directed along the positive z-axis:
and the negative sign is given by the right hand rule.
The cross product vector is:
The magnitude of the cross product is:
And, therefore, the area of the triangle formed by the vectors a and b can also be computed from their components:
The same computation can be done in 3D, to obtain a most useful formula for the area of a triangle in 3D.
we can reduce the cross product to:
Where from, the area of the 3D triangle formed by the vectors a and b is:
Example 1: Compute the area of the triangle with vertices at coordinates:
First, choose two of the sides of the triangle to be the operands for the cross-product:
Compute the components of the operands:
Compute the components of the cross product vector:
Compute the magnitude of the cross product vector:
Compute the area of the triangle: