N.VM

Vector and Matrix Quantities

N.VM.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).N.VM.10(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.N.VM.11(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.N.VM.12(+) Work with 22 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.N.VM.2(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.N.VM.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.N.VM.4(+) Add and subtract vectors.N.VM.4a(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.N.VM.4b(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.N.VM.4c(+) Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.N.VM.5(+) Multiply a vector by a scalar.N.VM.5a(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, (e.g. as c(v_x,v_y) = (cv_x, cv_y).)N.VM.5b(+) Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).N.VM.6(11) Use matrices to represent and manipulate data, (e.g. to represent payoffs or incidence relationships in a network.)N.VM.7(11) Multiply matrices by scalars to produce new matrices, (e.g. as when all of the payoffs in a game are doubled.)N.VM.8(11) Add, subtract, and multiply matrices of appropriate dimensions; find determinants of 22 matrices.N.VM.9(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
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