Alaska: Vector and Matrix Quantities Math Standards

(+) 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|,

(+) 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.

(+) 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.

(+) WORK WITH 2 × 2 MATRICES AS A TRANSFORMATIONS OF THE PLANE, AND INTERPRET THE ABSOLUTE VALUE OF THE DETERMINANT IN TERMS OF AREA.

(+) FIND THE COMPONENTS OF A VECTOR BY SUBTRACTING THE COORDINATES OF AN INITIAL POINT FROM THE COORDINATES OF A TERMINAL POINT.

(+) SOLVE PROBLEMS INVOLVING VELOCITY AND OTHER QUANTITIES THAT CAN BE REPRESENTED BY VECTORS.

(+) ADD AND SUBTRACT VECTORS.

• N-VM.4.a 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.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
• N-VM.4.c 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.

(+) MULTIPLY A VECTOR BY A SCALAR.

• N-VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
• N-VM.5.b 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).

(+) USE MATRICES TO REPRESENT AND MANIPULATE DATA, E.G., TO REPRESENT PAYOFFS OR INCIDENCE RELATIONSHIPS IN A NETWORK.

(+) MULTIPLY MATRICES BY SCALARS TO PRODUCE NEW MATRICES, E.G., AS WHEN ALL OF THE PAYOFFS IN A GAME ARE DOUBLED.

(+) ADD, SUBTRACT, AND MULTIPLY MATRICES OF APPROPRIATE DIMENSIONS.

(+) UNDERSTAND THAT, UNLIKE MULTIPLICATION OF NUMBERS, MATRIX MULTIPLICATION FOR SQUARE MATRICES IS NOT A COMMUTATIVE OPERATION, BUT STILL SATISFIES THE ASSOCIATIVE AND DISTRIBUTIVE PROPERTIES.