Trigonometry / Unit 8: Oblique Triangles and Vectors
8.6 Vector Operations and Applications
Notes
Add or subtract vectors component by component, and multiply by a scalar by scaling each component.
- $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$ and $k\langle a, b \rangle = \langle ka, kb \rangle$.
- Dot product: $\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$, a number. If it is 0, the vectors are perpendicular.
- The resultant is the sum of vectors acting together, such as a boat's velocity plus the current.
- Perpendicular vectors combine with the Pythagorean theorem.
Practice
Let $\vec{u} = \langle 2, -1 \rangle$ and $\vec{v} = \langle -3, 5 \rangle$. What is $2\vec{u} + \vec{v}$?
$2\vec{u} = \langle 4, -2 \rangle$, then add $\langle -3, 5 \rangle$.
Let $\vec{u} = \langle 3, 4 \rangle$ and $\vec{v} = \langle -4, 3 \rangle$. These vectors are:
$\vec{u} \cdot \vec{v} = -12 + 12 = 0$.
A boat heads due east at 12 mph in a current flowing due north at 5 mph. What is the boat's speed relative to the ground?
The velocities are perpendicular: $\displaystyle \sqrt{12^2 + 5^2} = 13$.
Ready for more? Take the Unit 8 test.