Precalculus Final Exam 2024

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.

Front

How do you find the magnitude of a vector \( \mathbf{v} = \langle x, y \rangle \)?

Back

The magnitude is calculated using the formula: \( ||\mathbf{v}|| = \sqrt{x^2 + y^2} \).

Front

What is the formula to find the angle \( \theta \) between two vectors?

Back

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

Front

How do you calculate the opposite direction of a vector \( \mathbf{a} \)?

Back

To find the unit vector in the opposite direction of \( \mathbf{a} \), first find the unit vector \( \hat{a} = \frac{\mathbf{a}}{||\mathbf{a}||} \), then multiply by -1: \( -\hat{a} \).

Front

What is the distance formula in a 2D coordinate system?

Back

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Front

What is the significance of the dot product of two vectors?

Back

The dot product measures the cosine of the angle between two vectors and indicates how much one vector extends in the direction of another.

Front

How do you scale a vector \( \mathbf{u} \) by a scalar \( k \)?

Back

To scale a vector \( \mathbf{u} = \langle x, y \rangle \) by a scalar \( k \), multiply each component by \( k \): \( k\mathbf{u} = \langle kx, ky \rangle \).

Front

What is the formula for the angle of elevation from a point to an object?

Back

The angle of elevation \( \theta \) can be found using: \( \tan(\theta) = \frac{h}{d} \), where \( h \) is the height of the object and \( d \) is the horizontal distance.

Front

How do you find the coordinates of a point given its distance from two fixed points?

Back

Use the distance formula to set up equations based on the distances from the fixed points, then solve the system of equations.

Front

What is the relationship between the sides of a right triangle and the angles?

Back

In a right triangle, the sides are related to the angles by the sine, cosine, and tangent functions: \( \sin(\theta) = \frac{opposite}{hypotenuse} \), \( \cos(\theta) = \frac{adjacent}{hypotenuse} \), \( \tan(\theta) = \frac{opposite}{adjacent} \).