The magnitude and direction of a vector tell how long the vector is and which way it points. For a vector ⟨a, b⟩, the magnitude is √(a² + b²). The direction angle θ is measured counterclockwise from the positive x-axis, with tan θ = b/a adjusted for the quadrant.
Identifying conic sections from an equation starts with the squared terms. Write the equation as Ax² + Cy² + Dx + Ey + F = 0. If A = C, it is a circle. If A and C have the same sign but differ, it is an ellipse. Opposite signs give a hyperbola, and only one squared term gives a parabola.
A hyperbola is a curve with two branches. Every point on it has the same difference of distances to two fixed points, the foci. The vertices are a units from the center, the foci are c units away with c² = a² + b², and two asymptotes guide the branches.
An ellipse is an oval: every point on it has the same total distance to two fixed points called foci. Its standard form is (x − h)²/a² + (y − k)²/b² = 1, with center (h, k). To graph it, plot the center, then the vertices a units away and the co-vertices b units away. The foci are c units from the center, where c² = a² − b².
Every point on a parabola is the same distance from a fixed point, the focus, and a fixed line, the directrix. To find the focus and directrix of a parabola, write its equation as (x − h)² = 4p(y − k) or (y − k)² = 4p(x − h). The focus is p units from the vertex, and the directrix is p units on the other side.
A matrix transformation moves points in the plane by multiplying each point, written as a column, by a 2 × 2 matrix. Different matrices rotate, reflect, stretch or shrink a figure. To transform a whole shape, put its vertices in the columns of one matrix and multiply once.
Cramer’s rule solves a system of linear equations with determinants. For each variable, replace that variable’s column of coefficients with the column of constants, find the determinant, and divide it by the determinant of the coefficient matrix. It works only when that main determinant is not 0.
The inverse of a matrix A, written A⁻¹, is the matrix that undoes A, so AA⁻¹ = I, the identity matrix. For a 2 × 2 matrix, swap the two main diagonal entries, change the signs of the other two, and divide by the determinant. A matrix has an inverse only when its determinant is not 0.
The determinant of a matrix is a single number calculated from a square matrix. For a 2 × 2 matrix with rows a, b and c, d, the determinant is ad − bc. For a 3 × 3 matrix, expand along a row using 2 × 2 determinants. A determinant of 0 means the matrix has no inverse.
To solve a system with an augmented matrix, write the coefficients and constants of the equations as one matrix, with a bar before the constants. Then use row operations (swap rows, multiply a row by a nonzero number, add a multiple of one row to another) to reach row-echelon form, and finish with back substitution, a method called Gaussian elimination.