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Selected answers to common 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
- 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 =
- 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
- 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
- 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
- Using Vectors in Geometry and Physics [07/10/1998]
How do you use vectors in problems about medians, areas, and acceleration
- 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...