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**Standard(s): **
[MA2015] PRE (9-12) 5 :

[MA2015] PRE (9-12) 6 :

[MA2015] PRE (9-12) 7 :

[MA2015] PRE (9-12) 8 :

[MA2015] PRE (9-12) 9 :

[MA2015] PRE (9-12) 10 :

5 ) (+) 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-VM1]

[MA2015] PRE (9-12) 6 :

6 ) (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. [N-VM2]

[MA2015] PRE (9-12) 7 :

7 ) (+) Solve problems involving velocity and other quantities that can be represented by vectors. [N-VM3]

[MA2015] PRE (9-12) 8 :

8 ) (+) Add and subtract vectors. [N-VM4]

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-VM4a]

b. (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. [N-VM4b]

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. [N-VM4c]

[MA2015] PRE (9-12) 9 :

9 ) (+) Multiply a vector by a scalar. [N-VM5]

a. (+) 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-VM5a]

b. (+) Compute the magnitude of a scalar multiple *c***v** using ||*c***v**|| = |*c*|v. Compute the direction of *c***v** knowing that when |*c*|v ≠ 0, the direction of *c***v** is either along **v** (for *c* > 0) or against **v** (for *c* < 0). [N-VM5b]

[MA2015] PRE (9-12) 10 :

10 ) (+) 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-VM11]

Module 2, Topic D opens with a formal definition of a vector. The arithmetical work for vector addition, subtraction, scalar multiplication, and vector magnitude is explored along with the geometrical frameworks for these operations. Students solve problems involving velocity and other quantities that can be represented by vectors. Parametric equations are introduced allowing students to connect their prior work with functions to vectors.