Vectors

Front

What are vectors?

Back

Vectors are quantities that have both magnitude and direction, represented graphically by arrows.

Front

What is the difference between parallel and orthogonal vectors?

Back

Parallel vectors point in the same or opposite directions, while orthogonal vectors are at right angles (90 degrees) to each other.

Front

How do you determine if two vectors are orthogonal?

Back

Two vectors are orthogonal if their dot product is zero.

Front

What is the formula for the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \)?

Back

The dot product is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Front

How do you find the angle between two vectors?

Back

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

Front

What is the component form of a vector?

Back

The component form of a vector \( \mathbf{v} \) in two dimensions is represented as \( \mathbf{v} = \langle v_x, v_y \rangle \), where \( v_x \) and \( v_y \) are the horizontal and vertical components.

Front

How do you convert a vector from magnitude and direction to component form?

Back

To convert a vector from magnitude \( r \) and angle \( \theta \) to component form: \( v_x = r \cos(\theta) \) and \( v_y = r \sin(\theta) \).

Front

What is the resultant vector?

Back

The resultant vector is the vector sum of two or more vectors, representing their combined effect.

Front

How do you find the resultant of two vectors?

Back

To find the resultant of two vectors, add their corresponding components: \( R_x = A_x + B_x \) and \( R_y = A_y + B_y \).

Front

What is the significance of the direction of a vector?

Back

The direction of a vector indicates where it is pointing, which is crucial for understanding its effect in physical applications.