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Browse High School Linear Algebra
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Vector questions.

Rotation Matrix Using Trig Functions [12/15/2005]
I know what the rotation matrix is for an angle of rotation of theta, but I don't understand why it works.

Rotations in Three Dimensions [11/11/1999]
How can I rotate something about an axis or about a line in 3D?

Row Operations and an Augmented Matrix [07/26/1999]
How can I solve a system of five simultaneous equations using row operations on the augmented matrix of the system?

Row-Reducing a Matrix [04/10/1998]
Could you please teach me how to row-reduce a matrix?

Shortest Distance between Points [01/17/1998]
I am doing a project on the shortest distance between two points via another plane. I need help with my theorems.

Shrunk Axes [01/20/1997]
How do you use the distance formula when the coordinate axes are not perpendicular?

Solve for x, y, and z [7/27/1996]
How do I solve for x, y, and z in these three equations: 1/2y + 1/3z = 26, 1/3x + 1/4z = 23, and 1/2x + 1/4y = 28?

Solving 3x3 Matrices Manually [01/27/1998]
How can I find the inverse of a 3x3 matrix manually (without pushing the inverse button on my calculator)?

Solving a 6x6 System of Equations [03/24/1998]
Is there a Cramer's rule for a 6x6 equation?

Solving a System More Swiftly [05/14/2017]
Given a system of seven equations in seven variables, an adult seeks an alternative to the conventional — and tedious — approach. Doctor Ian suggests a quicker way forward by exploiting common factors and a clever change of variables.

Solving Linear Equations in Computer Programs [02/13/2003]
What is the best and simplest method in computer programs for solving linear equations: Cramers method; Gaussian Elimination method; Inverse matrix method?

Solving Systems of Equations Using Matrices [04/02/1999]
Solve the following system of equations: a*(1-a) = b*c, b*(a+d-1) = 0, c*(a+d-1) = 0, d*(1-d) = b*c.

Solving Systems Using Augmented Matrices [11/05/1996]
How do you solve a set of linear equations using an augmented matrix?

Some Algebra Problems [6/1/1996]
If z=(3-2i)^1/2 then find z^-...

The Stationary Vector [11/13/2000]
In Markov processes, what does a stationary vector tell you and how do you find it?

Stretching and Shearing away from Coordinate Axes [08/27/2012]
A student struggles to calculate matrices that correspond to linear transformations that occur off of the x- or y-axes. Doctor Schwa formalizes what the student has already done, then introduces a second approach and the concept of conjugations.

Subspace [01/25/2001]
Determine whether the following is a subspace of R^3...

Sum of Two Vectors [9/10/1996]
Why is the magnitude of the sum of two vectors less than or equal to the sum of the magnitudes of each vector?

Symmetric Matrices [10/27/1999]
Find all values of a, b, and c for which a 3x3 matrix A is symmetric, where the elements of A contain algebraic expressions in a, b, and c.

Symmetric Matrices in Everyday Life [05/20/2013]
A student wonders where in the real world a symmetric matrix might come up. Doctor George offers up two examples.

System of Equations and Gauss-Jordan [11/29/1998]
Solve using the Gauss-Jordan method: a 5-percent solution of a drug is mixed with 15- and 10-percent solutions...

Three-Dimensional Cross Product Derivation [07/26/1999]
How can we prove the equation for 3-dimensional cross products in terms of the determinants a x (b x c) = (a.c)b - (a.b)c is correct?

Three-Dimensional Vectors [01/29/2003]
I am finding it very hard to understand and visualise the notion of a vector in 3 dimensions.

Two Dogs Pulling a Sled [10/15/2001]
What are the magnitude and direction of the force the person will have to exert on the sled to keep it moving along the centerline of the road at constant velocity?

Two Equations, Two Unknowns [7/27/1996]
How do I solve for x and y in terms of a and b for the equations x^4 + y^4 = b^4 and x + y = a?

Types of Linear Systems [10/13/1997]
What is meant when a type of linear system is said to be a consistent or an inconsistent system?

Understanding Vector Spaces [03/14/2001]
What does a vector space in R^n mean? How can I prove that a list of numbers is a vector space?

Unit Vectors [07/02/2002]
Find the two unit vectors that are collinear with each of the following vectors. (a) vector A = (3, -5). I don't understand--is a unit vector only ever equal to 1?

Using an Augmented Matrix [11/21/1999]
Solve the set of simultaneous equations x+3y+z=3, x+5y+5z=1 and 2x+6y+ 3z=8 using an augmented matrix?

Using Determinants to Find Area [06/09/1998]
How can you find the area of a triangle or a parallelogram using determinants?

Using Vectors in Geometry and Physics [07/10/1998]
How do you use vectors in problems about medians, areas, and acceleration and velocity?

Using Vectors to Prove Collinearity [06/05/1999]
How can I prove that the points of intersection of AB.DE, BC.EF, and CA.FD are collinear, if concurrent lines 0A, 0B, and 0C are produced to D, E, and F?

Values of Scalers [9/11/1996]
Can a scalar quantity be negative?

Variables Multiplied by Other Variables, Simplified Step by Step [08/22/2011]
If variables get multiplied by other variables in a system of equations, how do you solve it? or is it even possible? Doctor Vogler outlines several strategies, then -- reflecting on his choices at every turn -- methodically simplifies an example that requires lowering polynomial degree.

Vector Algebra: Finding the Intersection Point [04/23/2003]
If I have two lines in three dimensions that I know intersect at some point, how do I work out what that point is? Both lines are defined by two points on each line.

Vector Products and Possibilities [10/09/2012]
An engineer wonders how we come to define vector dot and cross products. Doctor Fenton derives the definition before Doctor Jacques reveals tensor, wedge, and other products.

Vector Proof [01/17/1999]
Prove that given P, Q, R, and S (any 4 non-collinear points), with A and B the midpoints of PR and QS respectively, then PQ + RS = 2 AB...

Vector Proof: Parallelogram Diagonals [01/20/1999]
Use vectors to prove that the diagonals of a parallelogram bisect each other and the line joining the midpoints of two sides of a triangle...

Vectors and Orientation [05/29/2003]
I am trying to write a game involving spaceships, and I have been having trouble finding relative positions.

Vectors of Parallelograms and Octagons [07/28/1998]
ABCDEFGH is a regular octagon and AB = p and BC = q. Express AH in terms of p and q...

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