Notes on Cartesian Vectors: Dot Product, Cross Product, and Projections

Cartesian Vectors in Cartesian Form

A vector in three dimensions can be written in Cartesian form using the three orthogonal unit vectors \mathbf{i}, \mathbf{j}, \mathbf{k}: each vector has components along x, y, z. If
\mathbf{a} = ax \mathbf{i} + ay \mathbf{j} + az \mathbf{k} and \mathbf{b} = bx \mathbf{i} + by \mathbf{j} + bz \mathbf{k}, then:

• Vector addition is performed componentwise:
  \n\mathbf{a} + \\mathbf{b} = (ax + bx) \mathbf{i} + (ay + by) \mathbf{j} + (az + bz) \mathbf{k}.
• Vector subtraction is also componentwise:
  \n\mathbf{b} - \mathbf{a} = (bx - ax) \mathbf{i} + (by - ay) \mathbf{j} + (bz - az) \mathbf{k}.

Dot Product: Definition, Geometry, and Basic Properties

The dot product of two Cartesian vectors is defined as

\mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz.

Geometrically, it also equals the product of the magnitudes and the cosine of the angle between them:

\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \, |\mathbf{b}| \, \cos \theta.

This makes the dot product useful for finding angles and projections.

• Utility: the dot product can be used to find the angle between two lines or vectors via
  \cos \theta = \\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \, |\mathbf{b}|}.
• Parallel and perpendicular components relative to a line: the dot product helps determine components along a given direction.
• The dot product is also called the scalar product, because the result is a scalar (not a vector).

Properties of the Dot Product

• Commutative law:
  
  \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.
  
• Distributive law:
  
  \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \cdot \mathbf{b}) + (\mathbf{a} \cdot \mathbf{c}).
  
• Scalar multiplication: for any scalar \lambda,
  (\\lambda \mathbf{a}) \cdot \mathbf{b} = \mathbf{a} \cdot (\\lambda \mathbf{b}) = \\lambda (\mathbf{a} \cdot \mathbf{b}).
  
• Unit vectors: \mathbf{
  3} = \hat{} \mathbf{i}, \mathbf{j}, \mathbf{k} play the role of the standard basis. Note that
  \mathbf{i} \cdot \mathbf{i} = 1, \mathbf{j} \cdot \mathbf{j} = 1, \mathbf{k} \cdot \mathbf{k} = 1,
  and any pair of distinct unit vectors are orthogonal:
  \mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{i} = \mathbf{i} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{j} = 0.
  

Projection and the Geometric Interpretation of the Dot Product

• The dot product provides the projection of one vector onto another. The projection of \mathbf{b} onto \mathbf{a} has length |\mathbf{b}| cos θ. Since a \cdot b = |a| |b| cos θ, it follows that the dot product equals the magnitude of the first vector times the projection of the second onto the first:
• Scalar projection of \mathbf{b} onto \mathbf{a}:
  a \cdot b = |\mathbf{a}| \, (\text{projection of } \mathbf{b} \text{ onto } \mathbf{a}).
• If we use the unit vector along \mathbf{a}, \hat{a} = \mathbf{a} / |\mathbf{a}|, then the vector projection of \mathbf{b} onto \hat{a} is
  \text{proj}_{\hat{a}} \\mathbf{b} = (\mathbf{b} \cdot \hat{a}) \hat{a}.
• In a force problem, the dot product helps quantify how much one force acts along the direction of another via projection.

Dot Product with the Basis Vectors and Components

• The dot product of any vector with the basis vectors gives its components:
  \,\mathbf{a} \cdot \mathbf{i} = ax, \mathbf{a} \cdot \mathbf{j} = ay, \mathbf{a} \cdot \mathbf{k} = a_z.
• For unit vectors interacting with themselves or with each other:
  \mathbf{i} \cdot \mathbf{i} = 1,
  \mathbf{j} \cdot \mathbf{j} = 1,
  \mathbf{k} \cdot \mathbf{k} = 1,
  
  \mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{i} = \mathbf{i} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{j} = 0.

Cross Product: Definition, Magnitude, and Direction

For two vectors \mathbf{a} and \mathbf{b}, the cross product is another vector:
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \cr ax & ay & az \cr bx & by & bz \end{vmatrix} = (ay bz - az by) \mathbf{i} + (az bx - ax bz) \mathbf{j} + (ax by - ay bx) \mathbf{k}.

• The cross product is a vector, not a scalar, and it is perpendicular to the plane containing \mathbf{a} and \mathbf{b}.
• Magnitude: |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \, |\mathbf{b}| \, \sin \theta.
• Direction is given by the right-hand rule: point the fingers from \mathbf{a} to \mathbf{b}, and the thumb points in the direction of \mathbf{a} \times \mathbf{b}.
• The cross product is not commutative:
  \mathbf{a} \times \mathbf{b}
  eq \mathbf{b} \times \mathbf{a}, \text{in fact } \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}).
• Scalar multiplication and distributivity:
  (\lambda \\mathbf{a}) \times \mathbf{b} = \lambda (\mathbf{a} \times \mathbf{b}) = \\mathbf{a} \times (\lambda \\mathbf{b}),
  \mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}.

Cross Product with Unit Vectors and Examples

• Fundamental cyclic relations (right-hand rule):
  \mathbf{i} \times \mathbf{j} = \mathbf{k}, \mathbf{j} \times \mathbf{k} = \mathbf{i}, \mathbf{k} \times \mathbf{i} = \mathbf{j}.
• Antisymmetry in reversed order:
  \mathbf{j} \times \mathbf{i} = -\mathbf{k}, \mathbf{k} \times \mathbf{j} = -\mathbf{i}, \mathbf{i} \times \mathbf{k} = -\mathbf{j}.
• Cross products of identical unit vectors vanish:
  \mathbf{i} \times \mathbf{i} = \mathbf{j} \times \mathbf{j} = \mathbf{k} \times \mathbf{k} = 0.

Cross Product in Components and Determinants

• A handy way to compute a \mathbf{a} \times \mathbf{b} is the determinant:
  \mathbf{a} \times \mathbf{b} = \det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \cr ax & ay & az \cr bx & by & bz \end{pmatrix} = (ay bz - az by) \mathbf{i} + (az bx - ax bz) \mathbf{j} + (ax by - ay bx) \mathbf{k}.

• Longhand (component-wise) expansion is consistent with the determinant method and is also shown in practice in lectures.

Projections, Components, and Three-Dimensional Geometry

• Projection of a vector onto a line (defined by a unit vector \hat{u}) can be expressed as:
  \text{Parallel component: } \mathbf{a}_{\parallel} = (\mathbf{a} \cdot \hat{u}) \hat{u}.
• The scalar projection (length along the line) is \mathbf{a} \cdot \hat{u}.
• The perpendicular component is the remainder: \mathbf{a}{\perp} = \mathbf{a} - \mathbf{a}{\parallel}.
• If you want the projection of a onto the axis defined by a non-unit vector, use
  \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \right) \mathbf{b}.
• For a unit axis along the x-direction, \hat{u} = \mathbf{i}, and the projection length is ax, while the projection vector is ax \mathbf{i}.
• In 3D, finding projections by pure geometry is more challenging; the dot product and unit vectors provide a straightforward method.
• The sign of the dot product indicates direction: if the result is positive, the projection aligns with the unit vector; if negative, it points opposite to that axis.

Practical Notes: Vector Operations in 3D Problems

• The resultant of scalar projections along axes can be used to analyze how one force projects onto another in a given direction.
• The dot product is a key tool for resolving forces and components along specified directions, especially when combining multiple forces or components in 3D space.
• Remember to maintain a consistent right-hand coordinate system throughout problems: in the standard system, \mathbf{i} points along +x, \mathbf{j} along +y, and \mathbf{k} along +z (positive z typically upwards).
• In all cross-products, watch for non-commutativity: swapping the order flips the sign of the result; in dot products the order does not matter.

Quick Reference Formulas

• Vectors in Cartesian form:
  \mathbf{a} = ax \mathbf{i} + ay \mathbf{j} + az \mathbf{k}, \quad \ \mathbf{b} = bx \mathbf{i} + by \mathbf{j} + bz \mathbf{k}.
• Sum and difference:
  \mathbf{a} + \mathbf{b} = (ax + bx) \mathbf{i} + (ay + by) \mathbf{j} + (az + bz) \mathbf{k}, \\mathbf{b} - \mathbf{a} = (bx - ax) \mathbf{i} + (by - ay) \mathbf{j} + (bz - az) \mathbf{k}.
• Dot product:
  \mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz = |\mathbf{a}| |\mathbf{b}| \, \cos \theta.
• Magnitude of a vector:
  |\mathbf{a}| = \sqrt{ax^2 + ay^2 + a_z^2}.
• Angle between vectors:
  \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \, |\mathbf{b}|}, \\theta = \arccos\left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \ |\mathbf{b}|} \right).
• Projection (vector) onto unit direction \hat{u}:
  \text{proj}_{\hat{u}} \mathbf{a} = (\mathbf{a} \cdot \hat{u}) \hat{u}.
• Cross product:
  \mathbf{a} \times \mathbf{b} = (ay bz - az by) \mathbf{i} + (az bx - ax bz) \mathbf{j} + (ax by - ay bx) \mathbf{k},
  |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta.
• Determinant form (alternative):
  \mathbf{a} \times \mathbf{b} = \det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \cr ax & ay & az \cr bx & by & bz \end{pmatrix}.
• Cyclic basis relations (right-hand rule):
  \mathbf{i} \times \mathbf{j} = \mathbf{k}, \mathbf{j} \times \mathbf{k} = \mathbf{i}, \mathbf{k} \times \mathbf{i} = \mathbf{j},
  and
  \mathbf{j} \times \mathbf{i} = -\mathbf{k}, \mathbf{k} \times \mathbf{j} = -\mathbf{i}, \mathbf{i} \times \mathbf{k} = -\mathbf{j}. $$
• Consistency reminder: always use the same right-hand coordinate system to avoid confusion across problems.