Pre-Calculus
GradesPre-Calculus & CalculusPre-CalculusVectors & Their Operations

Vectors & Their Operations

🔭 Pre-Calculus
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Vectors & Their Operations

Represent quantities with direction and magnitude, and compute dot products.

A vector has both magnitude and direction, unlike a scalar which has only magnitude. Vectors are represented as directed line segments or component form ⟨a, b⟩. Vector operations include addition, scalar multiplication, and the dot product.

v⃗=⟨a,b⟩∣v⃗∣=a2+b2\vec{v} = \langle a, b \rangle \qquad |\vec{v}| = \sqrt{a^2 + b^2}
u⃗+v⃗=⟨u1+v1,  u2+v2⟩cv⃗=⟨ca,cb⟩\vec{u} + \vec{v} = \langle u_1+v_1,\; u_2+v_2 \rangle \qquad c\vec{v} = \langle ca, cb \rangle
u⃗⋅v⃗=u1v1+u2v2=∣u⃗∣∣v⃗∣cos⁡θ\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 = |\vec{u}||\vec{v}|\cos\theta
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Two vectors are perpendicular (orthogonal) if and only if their dot product is zero: u⃗ · v⃗ = 0.

🎯Finding the Angle Between Vectors
u⃗ = ⟨3, 4⟩, v⃗ = ⟨1, 0⟩. Dot product = 3. |u⃗| = 5, |v⃗| = 1. cos θ = 3/(5·1) = 0.6. θ = arccos(0.6) ≈ 53.1°.
cos⁡θ=u⃗⋅v⃗∣u⃗∣∣v⃗∣=35  ⟹  θ=arccos⁡ ⁣(35)≈53.1∘\cos\theta = \frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|} = \frac{3}{5} \implies \theta = \arccos\!\left(\frac{3}{5}\right) \approx 53.1^\circ
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Remember This!

A unit vector has magnitude 1. To find the unit vector in the direction of v⃗, divide by its magnitude: û = v⃗/|v⃗|. Unit vectors are useful for expressing direction without magnitude.

✏️ Try It!

What is the dot product of u⃗ = ⟨2, −3⟩ and v⃗ = ⟨4, 1⟩?

Take Quiz 📝 — 25 Questions