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Three simple steps: (1) Share an assignment from thousands of standards-aligned topics using a share code. (2) Students speak and draw their thinking naturally, and Goblins responds with Socratic questioning that sounds just like their teacher. (3) Teachers get real-time formative insights showing what students struggle with and suggestions to inform instruction.

Does Goblins prevent cheating?

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Can I use my own voice with Goblins?

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What math levels does Goblins cover?

Goblins covers math from Grade 5 through AP Calculus, including thousands of standards-aligned topics. Teachers can also upload their own content for custom assignments at any level.

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Does Goblins support English Language Learners?

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What grade level is N-VM taught?

This standard is part of the N-VM curriculum in Alaska Mathematics Standards.

N-VM: Vector and Matrix Quantities

Vector and Matrix Quantities

Print

Vector and Matrix Quantities introduces vectors as quantities with both magnitude and direction, with operations including addition, scalar multiplication, and dot products. Matrices are defined and operated on, including addition, multiplication, and scalar multiplication, with attention to properties such as non-commutativity. Matrix methods connect to solving systems of equations and transformations in the plane.

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|,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 2 × 2 matrices as a 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.5(+) Multiply a vector by a scalar.N-VM.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.N-VM.7(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.N-VM.8(+) Add, subtract, and multiply matrices of appropriate dimensions.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.

Example Problems

These are the component forms of vectors

\vec{i}

and

\vec{j}

:

\vec{i} = (-6, 9)

\vec{j} = (3, 3)

Add the vectors.

Vector

\vec{AB}

has a terminal point

(-5, -5)

, an x component of 3, and a y component of 3.

Find the coordinates of the initial point, A.

Let b be a vector-valued function defined by

b(t) = (t^{-1}, 9^t)

.
Find

b'(t)

.

These are the component forms of vectors

\vec{w}

and

\vec{x}

:

\vec{w} = (2, 5)

\vec{x} = (-8, 9)

Subtract the vectors.

The component form of vector

\vec{v}

is

\vec{v} = (2, 0)

.
Find

5\vec{v}

.

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