4.3: Vectors | AP Precalculus | Code Scholars Tutoring

How it works

A vector records magnitude and direction. Add components to combine displacements; multiply all components to scale. A position vector and a point may use the same coordinate pair, but they represent different objects.

\mathbf{u}+\mathbf{v}=(u_1+v_1,u_2+v_2),\quad\|\mathbf{v}\|=\sqrt{v_1^2+v_2^2}

Worked examples

Example 1

Find the vector from $\displaystyle A=(1,2)$ to $\displaystyle B=(4,6)$ and its magnitude.

Subtract coordinates: $\displaystyle \overrightarrow{AB}=(3,4)$ , whose magnitude is $\displaystyle 5$ .

Example 2

For $\displaystyle \mathbf{u}=(2,-1)$ and $\displaystyle \mathbf{v}=(3,4)$ , find $\displaystyle 2\mathbf{u}+\mathbf{v}$ .

The result is $\displaystyle (4,-2)+(3,4)=(7,2)$ .

Problems

Problem 4.3.1

A displacement is $\displaystyle \mathbf{u}=(3,1)$ followed by $\displaystyle \mathbf{v}=(1,2)$ . Find the resultant and its magnitude.

Vector 3,1 from the origin followed by vector 1,2 from its tip. — Place the second vector at the tip of the first.

Show worked solution

The resultant is $\displaystyle (4,3)$ with magnitude $\displaystyle 5$ .

Vector 3,1 from the origin followed by vector 1,2 from its tip. The solution adds labeled relationships. — The resultant from the origin is (4,3), with magnitude 5.

Problem 4.3.2

Find a unit vector in the direction $\displaystyle (6,8)$ .

Show worked solution

Its magnitude is $\displaystyle 10$ , so a unit vector is $\displaystyle (3/5,4/5)$ .

Problem 4.3.3

Compute $\displaystyle (2,1)\cdot(-1,2)$ and interpret the result.

Show worked solution

The dot product is $\displaystyle -2+2=0$ , so the nonzero vectors are perpendicular.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.3.1

Problem 4.3.2

Problem 4.3.3

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