Key takeaway: This precalculus vectors lesson shows how to describe a quantity that has both size and direction. A vector in component form is \(\langle a,b\rangle\), and its magnitude is \(\sqrt{a^2+b^2}\). You will add, subtract and scale vectors, then use the dot product to find angles. The page follows N-VM.A.1 and N-VM.B.4, and it ends with boats, planes and forces.
A vector is a quantity with magnitude and direction, written in component form as ⟨a, b⟩. Its magnitude is the square root of a squared plus b squared. Vectors add, subtract, and scale component by component, and the dot product u·v equals |u||v| cos θ, which gives angles and tests for perpendicular vectors.
Precalculus Vectors Lesson: key facts
- Magnitude of ⟨a, b⟩ is √(a² + b²); direction angle satisfies tan θ = b/a in the correct quadrant.
- Add and subtract component by component; the parallelogram rule gives the same sum geometrically.
- Scalar multiplication k⟨a, b⟩ = ⟨ka, kb⟩ changes the length by |k| and reverses direction when k is negative.
- A unit vector in the direction of v is v/|v|; every vector is a·i + b·j.
- u·v = u₁v₁ + u₂v₂ = |u||v| cos θ, and u·v = 0 exactly when u and v are perpendicular.
What is a vector, and how do you find its magnitude and direction?
A vector is a quantity with both a size and a direction, drawn as an arrow. In component form \(\vec{v}=\langle a, b\rangle\) the number \(a\) is the horizontal change and \(b\) is the vertical change. The magnitude is \(|\vec{v}|=\sqrt{a^{2}+b^{2}}\). The direction angle \(\theta\) is measured counterclockwise from the positive x-axis, and \(\tan\theta=\frac{b}{a}\) once you place the angle in the correct quadrant. For \(\vec{v}=\langle 3, 4\rangle\), the magnitude is \(\sqrt{9+16}=5\) and \(\theta=\tan^{-1}(\frac{4}{3})\approx 53.1^{\circ}\).
Unit vector: a vector of magnitude 1. The unit vector in the direction of \(\vec{v}\) is \(\frac{\vec{v}}{|\vec{v}|}\).
Standard unit vectors: \(\vec{i}=\langle 1, 0\rangle\) and \(\vec{j}=\langle 0, 1\rangle\), so \(\langle a, b\rangle=a\vec{i}+b\vec{j}\).
Components from magnitude and angle: \(\langle r\cos\theta,\ r\sin\theta\rangle\).
How do you add, subtract, and scale vectors?
Add or subtract vectors one component at a time: \(\langle a, b\rangle+\langle c, d\rangle=\langle a+c,\ b+d\rangle\). Geometrically, place the tail of the second arrow at the tip of the first, or draw both from one point and complete a parallelogram. The diagonal from the shared tail is the sum. To subtract, add the opposite: \(\vec{u}-\vec{v}=\vec{u}+(-\vec{v})\).
A scalar is an ordinary number. Multiplying by a scalar \(k\) scales every component: \(k\langle a, b\rangle=\langle ka, kb\rangle\). The length changes by the factor \(|k|\), and a negative \(k\) reverses the direction.
What do the dot product and the angle between vectors tell you?
The dot product of \(\vec{u}=\langle u_{1}, u_{2}\rangle\) and \(\vec{v}=\langle v_{1}, v_{2}\rangle\) is the number \(\vec{u}\cdot\vec{v}=u_{1}v_{1}+u_{2}v_{2}\). It also equals \(|\vec{u}||\vec{v}|\cos\theta\), so \(\cos\theta=\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\). A dot product of 0 means the vectors are perpendicular. For \(\langle 3, 1\rangle\) and \(\langle 1, 3\rangle\): \(\cos\theta=\frac{6}{\sqrt{10}\sqrt{10}}=0.6\), so \(\theta\approx 53.1^{\circ}\).
How do you solve velocity and force problems?
- Choose axes and write each quantity as a vector in component form.
- Add the vectors to get the resultant.
- Find its magnitude and direction angle.
- State the answer with units.
Example. A boat heads north across a river at 6 mph while the current flows east at 2.5 mph. The ground velocity is \(\langle 0, 6\rangle+\langle 2.5, 0\rangle=\langle 2.5, 6\rangle\). Its speed is \(\sqrt{2.5^{2}+6^{2}}=6.5\) mph, and its direction is \(\tan^{-1}(\frac{6}{2.5})\approx 67.4^{\circ}\) above east, which is about \(22.6^{\circ}\) east of north.
Vectors in the Plane: frequently asked questions
What is the difference between a vector and a scalar?
A scalar is a single number such as 12 mph or 5 lb. A vector also has a direction, such as 12 mph heading northeast. Speed is a scalar; velocity is a vector.
How do you find the direction angle of a vector in the second or third quadrant?
Compute the reference angle with the inverse tangent of the absolute value of b/a, then adjust for the quadrant. In quadrant II the angle is 180° minus the reference angle; in quadrant III it is 180° plus the reference angle. Many calculators handle this with a two-argument arctangent.
Why is the dot product a number and not a vector?
The dot product multiplies matching components and adds the results, so everything collapses to a single scalar. It measures how much two vectors point in the same direction.
How do you know two vectors are parallel?
Two nonzero vectors are parallel when one is a scalar multiple of the other. For example, ⟨2, 3⟩ and ⟨6, 9⟩ are parallel because the second is 3 times the first.
How do you add two vectors given by magnitude and angle?
Convert each vector to components with r cos θ and r sin θ, add the x-components and the y-components, and then convert the resultant back to a magnitude and direction angle.
How do you find the magnitude and direction of a vector?
For \(\langle-1,1\rangle\), the magnitude is \(\sqrt{2}\). The reference angle is \(45^{\circ}\), but the point lies in Quadrant II. Therefore the direction angle is \(135^{\circ}\), not \(-45^{\circ}\). A quick sketch prevents this common slip.
How do you add and scale vectors in a precalculus vectors lesson?
Work component by component. For example, \(3\langle2,-1\rangle-\langle1,4\rangle\) becomes \(\langle6,-3\rangle-\langle1,4\rangle=\langle5,-7\rangle\). Geometrically, the parallelogram rule gives the same sum. Multiplying by a negative number reverses the direction, while the length scales by the absolute value.
What does the dot product tell you about an angle?
The formula \(\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta\) links components to the angle. When the dot product is zero, the vectors are perpendicular. For \(\langle2,3\rangle\) and \(\langle3,-2\rangle\), the product is \(6-6=0\). Notice that the result is a number, not a vector.
How do you solve a boat or plane problem?
Treat each motion as a vector and add them. A boat heading straight across at 12 mph in a 5 mph current has ground speed \(\sqrt{12^2+5^2}=13\) mph. In contrast, the two speeds do not simply add, because they point in perpendicular directions.
Which skills should you rehearse after this vector lesson?
Sketch ten vectors and find each direction angle without a calculator. Then verify two perpendicular pairs with the dot product. Finally, write a short word problem about wind or current, and solve it in components.
See every chapter on the Precalculus math page.
