- Level: International Baccalaureate
- Subject: Maths
- Word count: 3020
Matrices and Vectors Summary
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Introduction
Rishi Garg - Page
Matrices
Matrix – a way of presenting information in table-like form:
This matrix has two rows and two columns, so it is said to be a matrix of order 2 x 2. An n x m matrix would have n rows and m columns. Each number in a matrix is called an element.
Operations with matrices – When adding and subtracting, each element is added to or subtracted from its corresponding element in the other matrix. For example:
Addition and subtraction can only be done with matrices of the same order. This is because if matrices of different orders were used, some elements would not have another corresponding element to be added to or subtracted from.
When multiplying matrices, the top-left element in the first matrix is multiplied by the top-left element in the second matrix. The product is then added to the product of the second element in the first row of the first matrix and the first element in the second row of the second matrix. This continues in a similar fashion for the rest of the elements. For example:
Because of this process, matrices can only be multiplied together if the number of columns in the first matrix equals the number of rows in the second matrix.
The additive identity is a matrix with all elements zero. The additive inverse is a matrix with elements such that when added to another matrix, the resulting matrix has all zeros.
Determinant – the determinant of a 2 x 2 matrix is obtained as follows:
The first three portions of the equation are simply different ways to notate “determinant of matrix A”. To find the determinant of a 3 x 3 matrix, do the following:
Middle
More advanced techniques
Find the angle between two vectors – use the two forms of the dot product to solve for Θ. For example, consider vectors and :
Find the dot product of the two vectors using the first form of the dot product: | |
Plug the dot product into the second form, along with the magnitudes of the two vectors: | |
Solve for Θ: |
Write a vector in terms of two other vectors – because vectors are not pinned down to any one location, any vector can be written in terms of any two other vectors. Consider, for example, the vector . In order to be written in terms of component vectors and , the following must be done:
Write the three vectors in a form that allows the component vectors to be multiplied by a scalar: | |
Now λ and μ must be found, so that the component vectors can be multiplied by the correct scalars. In order to do this, write the equation as a system of equations: | |
Solve the system of equations (using any method): | |
Now it is known what the values of λ and μ are. Simply multiply the component vectors by these values and then add the resulting vectors in order to obtain the original vector. |
Lines
Vector equation – a straight line is uniquely defined if given the position of one point that lies on the line and the direction of the line. In terms of vectors, this means that a line is defined if the following is known:
- The position vector of any point on the line
- Any vector that is parallel to the line
Therefore, the vector equation of a line is written as follows:
Where a (position vector) is the position vector of any point on the line, b (direction vector) is any vector parallel to the line, and r
Conclusion
Position vector a and vector d form a right triangle. Make the angle between them (at the origin) be θ. This means that:
Recall that the alternate form of the dot product is:
Using this form of the dot product and our previous equation, it can be proven that:
Finally:
Where d symbolizes the magnitude (or length) of vector d.
For example, the distance from plane with scalar product formula to the origin is .
Find distance from plane to plane – the easiest way to find this distance is to simply find the distance from the origin to each of the planes, and then subtract the distance. For example, taking the plane from the previous example and adding the plane , the distance between them is:
Note that this method only works with parallel planes, because two non-parallel planes intersect and the distance between them is zero.
Find distance from plane to line – the easiest way to find this distance is to pretend that the line is a plane, and then use the method described in the previous paragraph. For example, considering the plane and the line , the line can be changed into a parallel plane as follows:
Now, the distance between the original plane and the new “plane” can be found easily:
Find distance from plane to point – this method is almost exactly the same as the previous method. Simply change the point into a plane that is parallel to the original plane as shown in the above example, then find the distance between the two planes.
Find line intersection of two planes – take, for example, the planes 7x-4y+3z=-3 and 4x+2y+z=4:
Eliminate z from both equations (system of equations): | Or Or |
Eliminate y from both of the original equations: | Or |
Therefore, the Cartesian equation of the intersecting line is: |
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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