Front

What is a vector?

Back

A vector is a quantity that has both magnitude and direction, represented graphically by an arrow.

Front

How do you find the angle between two vectors?

Back

Use the formula: \( \theta = \cos^{-1}\left( \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \right) \) where \( \mathbf{u} \cdot \mathbf{v} \) is the dot product of the vectors.

Front

What is a unit vector?

Back

A unit vector is a vector with a magnitude of 1, indicating direction only.

Front

How do you find the unit vector in the direction of vector \( \mathbf{v} \)?

Back

Divide the vector \( \mathbf{v} \) by its magnitude: \( \hat{v} = \frac{\mathbf{v}}{||\mathbf{v}||} \).

Front

What is the component form of a vector?

Back

The component form of a vector \( \mathbf{v} \) from point A(x1, y1) to point B(x2, y2) is given by \( \mathbf{v} = \langle x2 - x1, y2 - y1 \rangle \).

Front

What does it mean for two vectors to be parallel?

Back

Two vectors are parallel if they have the same or opposite direction, which can be determined if one is a scalar multiple of the other.

Front

What does it mean for two vectors to be orthogonal?

Back

Two vectors are orthogonal if their dot product is zero, indicating they are at right angles to each other.

Front

How do you determine if two vectors are parallel, orthogonal, or neither?

Back

Calculate the dot product and check for scalar multiples. If the dot product is zero, they are orthogonal; if one vector is a scalar multiple of the other, they are parallel.

Front

What is the dot product of two vectors \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \)?

Back

The dot product is given by \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).

Front

What is the formula for the magnitude of a vector \( \mathbf{v} = \langle x, y \rangle \)?

Back

The magnitude is given by \( ||\mathbf{v}|| = \sqrt{x^2 + y^2} \).