Vectors

Front

What are the vertical and horizontal components of a vector?

Back

The vertical and horizontal components of a vector can be found using trigonometric functions. For a vector represented as \( \mathbf{v} = \langle x, y \rangle \), the horizontal component is \( x \) and the vertical component is \( y \).

Tags

CCSS.HSN.VM.A.2

Front

How do you add two vectors?

Back

To add two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), you add their corresponding components: \( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \).

Tags

CCSS.HSN-VM.B.4A

CCSS.HSN-VM.B.4B

Front

What is the formula to find the direction angle of a vector?

Back

The direction angle \( \theta \) of a vector \( \mathbf{v} = \langle x, y \rangle \) can be found using the formula: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

Tags

CCSS.HSN.VM.A.1

Front

What is the significance of the angle in vector representation?

Back

The angle of a vector indicates its direction in the coordinate system. It helps in understanding how the vector is oriented relative to the axes.

Tags

CCSS.HSN.VM.A.1

Front

What is the relationship between the components of a vector and its magnitude?

Back

The magnitude \( ||\mathbf{v}|| \) of a vector \( \mathbf{v} = \langle x, y \rangle \) is given by the formula: \( ||\mathbf{v}|| = \sqrt{x^2 + y^2} \).

Tags

CCSS.HSN.VM.A.1

Front

How do you determine if two vectors are equal?

Back

Two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) are equal if and only if their corresponding components are equal: \( u_1 = v_1 \) and \( u_2 = v_2 \).

Tags

CCSS.HSN.VM.A.1

Front

What is a unit vector?

Back

A unit vector is a vector that has a magnitude of 1. It is often used to indicate direction. A unit vector in the direction of vector \( \mathbf{v} \) is given by \( \hat{v} = \frac{\mathbf{v}}{||\mathbf{v}||} \).

Tags

CCSS.HSN.VM.A.1

Front

How do you find the dot product of two vectors?

Back

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is calculated as: \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \).

Front

What is the cross product of two vectors?

Back

The cross product of two vectors in three-dimensional space \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) results in a vector that is perpendicular to both: \( \mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle \).

Front

What is the angle between two vectors?

Back

The angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \) can be found using the formula: \( \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \).