Mathematics » High School: Number & Quantity » Vector & Matrix Quantities

Represent and model with vector quantities.

• 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).

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• 2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

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• 3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

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Perform operations on vectors.

• 4. (+) Add and subtract vectors.

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  • 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.

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  • Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

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  • 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.

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• 5. (+) Multiply a vector by a scalar.

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  • 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).

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  • 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).

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Perform operations on matrices and use matrices in applications.

• 6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

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• 7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

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• 8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

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• 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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• 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.

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• 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.

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• 12. (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

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