The Main Idea 

Vectors show up constantly in real life—think about a boat crossing a river (engine pushing one way, current pushing another) or a quarterback throwing a football (force and angle combined). To solve these problems, you need to know how to combine vectors.

Scalar Multiplication: Stretching and Flipping

When you multiply a vector [latex]\mathbf{v}[/latex] by a scalar [latex]k[/latex], you change its length and possibly its direction:

• If [latex]k > 0[/latex]: The vector stretches or shrinks but keeps the same direction. For example, [latex]2\mathbf{v}[/latex] doubles the length; [latex]0.5\mathbf{v}[/latex] halves it.
• If [latex]k < 0[/latex]: The vector flips to point the opposite direction. For instance, [latex]-\mathbf{v}[/latex] has the same length as [latex]\mathbf{v}[/latex] but points the other way.
• If [latex]k = 0[/latex]: You get the zero vector [latex]\mathbf{0}[/latex].

Vector Addition: Two Methods, Same Result

To add vectors [latex]\mathbf{v} + \mathbf{w}[/latex], use either approach:

• Triangle Method: Place the starting point of [latex]\mathbf{w}[/latex] at the endpoint of [latex]\mathbf{v}[/latex]. The sum runs from [latex]\mathbf{v}[/latex]‘s start to [latex]\mathbf{w}[/latex]‘s end. Think of it like walking from A to B, then B to C—your total displacement goes directly from A to C.
• Parallelogram Method: Start both vectors at the same point and complete the parallelogram. The diagonal is your sum.

Order doesn’t matter: [latex]\mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v}[/latex] (commutative property).

Triangle Inequality: The sum’s length satisfies [latex]||\mathbf{v} + \mathbf{w}|| \leq ||\mathbf{v}|| + ||\mathbf{w}||[/latex]. Equality only happens when vectors point the same direction.

Vector Subtraction: Adding the Opposite

For [latex]\mathbf{v} - \mathbf{w}[/latex], rewrite it as [latex]\mathbf{v} + (-\mathbf{w})[/latex]. Graphically, the difference points from [latex]\mathbf{w}[/latex]‘s endpoint to [latex]\mathbf{v}[/latex]‘s endpoint.

The Main Idea 

The dot product is a way to “multiply” two vectors, but the result is a scalar (a number), not another vector. It measures how much two vectors align with each other and is essential for calculating work, finding angles between vectors, and determining if vectors are perpendicular.

Calculating the Dot Product

For vectors [latex]\mathbf{u} = \langle u_1, u_2, u_3 \rangle[/latex] and [latex]\mathbf{v} = \langle v_1, v_2, v_3 \rangle[/latex]:

[latex]\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3[/latex]

Process: Multiply corresponding components, then add the results.

For 2D vectors [latex]\mathbf{u} = \langle u_1, u_2 \rangle[/latex] and [latex]\mathbf{v} = \langle v_1, v_2 \rangle[/latex], it’s the same: [latex]\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2[/latex].

Example: [latex]\langle 3, 5, 2 \rangle \cdot \langle -1, 3, 0 \rangle = (3)(-1) + (5)(3) + (2)(0) = -3 + 15 + 0 = 12[/latex]

Key Properties

• Commutative: [latex]\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}[/latex]
• Distributive: [latex]\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}[/latex]
• Magnitude relationship: [latex]\mathbf{v} \cdot \mathbf{v} = ||\mathbf{v}||^2[/latex]

Important Observations

• [latex]\mathbf{0} \cdot \mathbf{v} = 0[/latex] for any vector
• A vector’s dot product with itself equals its magnitude squared: [latex]\mathbf{v} \cdot \mathbf{v} = ||\mathbf{v}||^2[/latex]
• Only the zero vector has a dot product of zero with itself

The dot product tells you how much two vectors point in the same direction. It’s the foundation for calculating work (force in the direction of motion) and finding angles between vectors.

The Main Idea 

The dot product has a geometric interpretation that connects it to angles between vectors. This makes it a powerful tool for finding angles and determining if vectors are perpendicular.

Geometric Dot Product Formula

For nonzero vectors [latex]\mathbf{u}[/latex] and [latex]\mathbf{v}[/latex] with angle [latex]\theta[/latex] between them:

[latex]\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| , ||\mathbf{v}|| \cos \theta[/latex]

Rearranging to solve for the angle:

[latex]\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| , ||\mathbf{v}||}[/latex]

Then [latex]\theta = \arccos\left(\frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| , ||\mathbf{v}||}\right)[/latex]

What the Angle Tells You

The sign of [latex]\cos \theta[/latex] reveals the relationship between vectors:

• [latex]\cos \theta > 0[/latex]: Acute angle (vectors point in similar directions)
• [latex]\cos \theta = 0[/latex]: Right angle (vectors are perpendicular)
• [latex]\cos \theta < 0[/latex]: Obtuse angle (vectors point in opposite-ish directions)

Orthogonal Vectors: The Perpendicular Test

Nonzero vectors [latex]\mathbf{u}[/latex] and [latex]\mathbf{v}[/latex] are orthogonal (perpendicular) if and only if [latex]\mathbf{u} \cdot \mathbf{v} = 0[/latex].

To test if vectors are perpendicular, just compute their dot product. If it’s zero, they’re orthogonal.

Direction Angles and Cosines

A vector’s direction angles ([latex]\alpha[/latex], [latex]\beta[/latex], [latex]\gamma[/latex]) are the angles it makes with the positive [latex]x[/latex]-, [latex]y[/latex]-, and [latex]z[/latex]-axes. Find them by taking dot products with [latex]\mathbf{i}[/latex], [latex]\mathbf{j}[/latex], and [latex]\mathbf{k}[/latex].