Vectors #
Magnitude #
The magnitude of a vector is denoted by double bars, like \( ||\vec{a}|| \). It represents the length of the vector, which can be found using the Pythagorean theorem.
For example, the magnitude of the vector \( [3, 4] \) is \( \sqrt{3^2 + 4^2} = 5 \).
Exercise 1: What is the magnitude of vector \( [5, 12] \)?
Exercise 2: What is the product of \( ||\vec{a}|| \) and \( ||\vec{b}|| \) if \( a = [1, 2] \) and \( b = [2, 3] \)?
The Dot Product #
The dot product of two vectors is the sum of the products of corresponding components. For example, for vectors \( [1, 2, 3] \) and \( [4, 5, 6] \), the dot product is
\( (1 \cdot 4) + (2 \cdot 5) + (3 \cdot 6) = 32 \),
which can be written as \( [1, 2, 3] \cdot [4, 5, 6] = 32 \).
- If the dot product is zero, the vectors are perpendicular.
- If it’s positive, the angle between them is acute.
- If it’s negative, the angle is obtuse.
Exercises #
Exercise 3: What is the value of \( [1, 2] \cdot [3, 4] \)?
Exercise 4: Do the vectors \( [7, 8] \) and \( [9, 10] \) form an acute, right, or obtuse angle?
The Dot Product Equation #
The dot product formula can also be expressed as:
where:
- \( D \) is the dot product,
- \( M \) is the product of the magnitudes, and
- \( \theta \) is the angle between the vectors.
Example: Finding the Angle Between Two Vectors #
Find the angle between \( [1, 2] \) and \( [3, 4] \):
\[ \begin{aligned} D &= M \cos(\theta) \\ (1 \cdot 3) + (2 \cdot 4) &= ||[1,2]|| \cdot ||[3,4]|| \cdot \cos(\theta) \\ 11 &= 5\sqrt{5} \cdot \cos(\theta) \\ \frac{11}{5\sqrt{5}} &= \cos(\theta) \\ \arccos\left(\frac{11}{5\sqrt{5}}\right) &= \theta \\ \theta &\approx 79.6^\circ \end{aligned} \]Example: Solving for a Missing Component #
Let vectors \( [1, A] \) and \( [3, 4] \) have a dot product of 10 and form a \( 45^\circ \) angle. What is the value of \( A \)?
\[ \begin{aligned} D &= M \cos(\theta) \\ 10 &= 5 \cdot \sqrt{1^2 + A^2} \cdot \cos(45^\circ) \\ 10 &= 5 \cdot \sqrt{1 + A^2} \cdot \frac{\sqrt{2}}{2} \\ 2 &= \sqrt{1 + A^2} \cdot \frac{\sqrt{2}}{2} \\ 4 &= \sqrt{1 + A^2} \cdot \sqrt{2} \\ \frac{4}{\sqrt{2}} &= \sqrt{1 + A^2} \\ 8 &= 1 + A^2 \\ A^2 &= 7 \\ A &= \pm \sqrt{7} \end{aligned} \]Example: Vector Rotation #
Let \( \vec{p} = [1, 2] \) and suppose it’s rotated counterclockwise by \( 56^\circ \), resulting in a new vector \( \vec{r} = [A, B] \) such that the dot product is 30. What is the new vector?
\[ \begin{aligned} D &= M \cos(\theta) \\ 30 &= \sqrt{5} \cdot \sqrt{A^2 + B^2} \cdot \cos(56^\circ) \\ 30 &= \sqrt{5A^2 + 5B^2} \cdot \cos(56^\circ) \\ \frac{30}{\cos(56^\circ)} &= \sqrt{5A^2 + 5B^2} \\ \frac{900}{\cos^2(56^\circ)} &= 5A^2 + 5B^2 \end{aligned} \]Also, we know that:
\[ A + 2B = 30 \]This creates the system:
\[ \left\{ \begin{aligned} 5A^2 + 5B^2 &= \frac{900}{\cos^2(56^\circ)} \\ A + 2B &= 30 \end{aligned} \right. \]Solving this system yields the approximate solutions:
- \( [-11.79, 20.90] \)
- \( [23.80, 3.10] \)
Only the first is a counterclockwise rotation, so the final answer is approximately
\( \vec{r} \approx [-11.79, 20.90] \)
Exercises #
Exercise 5: What is the angle between the vectors \( [7, 8] \) and \( [2, 9] \)?
Exercise 6: What is the angle between the vectors \( [4, 2] \) and \( [8, 4] \)? Why does this operation lead to an error?
Exercise 7: What is the angle between the vectors \( [4, 2] \) and \( [2x, x] \)?
Exercise 8: Let \( \vec{a} = [3, 5] \), and let \( \vec{b} \) be the result when \( \vec{a} \) is rotated 30 degrees. What is the sum of the components of \( \vec{b} \)? Round to the nearest tenth.
Exercise 9: What is the angle between the lines \( 5x + 4y = 9 \) and \( 8x + 3y = 5 \)?
Credits #
Explanations and problems by Adrian Hernandez Vega.