2.3 The Dot Product

Vector operations are crucial in understanding 3D space. They help us calculate distances, angles, and relationships between objects. Dot products, projections, and direction cosines are key tools for analyzing vector interactions.

These concepts have wide-ranging applications in physics and engineering. From calculating work done by forces to determining perpendicularity, vector operations provide a powerful framework for solving real-world problems in three dimensions.

Vector Operations and Applications

Dot product calculation and interpretation

Top images from around the web for Dot product calculation and interpretation

• 

  geometry - Dot Product Intuition - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Representation of dot product of a vector - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  geometry - Dot Product Intuition - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Representation of dot product of a vector - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Dot product calculation and interpretation

• 

  geometry - Dot Product Intuition - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Representation of dot product of a vector - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Dot product - Wikipedia View original

  Is this image relevant?

• 

  geometry - Dot Product Intuition - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Representation of dot product of a vector - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Calculates the dot product of two vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$ by multiplying corresponding components and summing the results: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
• Geometrically interprets the dot product as the product of the vector magnitudes and the cosine of the angle between them: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
  • Example: If $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 4, 5, 6 \rangle$, then $\vec{a} \cdot \vec{b} = 1(4) + 2(5) + 3(6) = 32$
• Results in a scalar value, not a vector, representing the degree to which the vectors are aligned
  • Positive dot product indicates vectors pointing in similar directions (acute angle)
  • Negative dot product indicates vectors pointing in opposite directions (obtuse angle)
• Also known as the inner product in linear algebra and functional analysis

Vector perpendicularity via dot product

• Determines if two vectors are perpendicular (orthogonal) by checking if their dot product is zero: $\vec{a} \cdot \vec{b} = 0$ if and only if $\vec{a} \perp \vec{b}$
  • Example: $\vec{a} = \langle 1, 0, 0 \rangle$ and $\vec{b} = \langle 0, 1, 0 \rangle$ are perpendicular since $\vec{a} \cdot \vec{b} = 1(0) + 0(1) + 0(0) = 0$
• Follows from the geometric interpretation of the dot product: if $\vec{a} \perp \vec{b}$, then $\cos\theta = 0$ (since $\theta = 90^\circ$), and thus $\vec{a} \cdot \vec{b} = 0$
  • Useful for finding orthogonal vectors or verifying perpendicularity in 3D space

Direction cosines and significance

• Computes the direction cosines of a vector $\vec{a} = \langle a_1, a_2, a_3 \rangle$ as the cosines of the angles between the vector and the positive x, y, and z axes: $\cos\alpha = \frac{a_1}{|\vec{a}|}$, $\cos\beta = \frac{a_2}{|\vec{a}|}$, and $\cos\gamma = \frac{a_3}{|\vec{a}|}$
  • Example: For $\vec{a} = \langle 1, 2, 2 \rangle$, $|\vec{a}| = \sqrt{1^2 + 2^2 + 2^2} = 3$, so $\cos\alpha = \frac{1}{3}$, $\cos\beta = \frac{2}{3}$, and $\cos\gamma = \frac{2}{3}$
• Describes the orientation of a vector in 3D space relative to the coordinate axes
• Satisfies the property that the sum of the squares of the direction cosines is always 1: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$
  • Helps normalize vectors and ensures consistency in vector orientation representation

Vector projections explained

• Calculates the projection of a vector $\vec{a}$ onto a vector $\vec{b}$ as the component of $\vec{a}$ that is parallel to $\vec{b}$: $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\frac{\vec{b}}{|\vec{b}|} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$
  • Example: If $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 1, 0, 0 \rangle$, then $\text{proj}_{\vec{b}}\vec{a} = \frac{1(1) + 2(0) + 3(0)}{1^2 + 0^2 + 0^2}\langle 1, 0, 0 \rangle = \langle 1, 0, 0 \rangle$
• Results in a vector with the same direction as $\vec{b}$ (or opposite if $\vec{a} \cdot \vec{b} < 0$)
• Has a magnitude equal to $|\text{proj}_{\vec{b}}\vec{a}| = |\vec{a}|\cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
  • Useful for decomposing vectors into parallel and perpendicular components

Dot product in work problems

• Applies the dot product to calculate work in physics as the product of the force vector $\vec{F}$ and the displacement vector $\vec{d}$: $W = \vec{F} \cdot \vec{d}$
  • Example: If a force $\vec{F} = \langle 2, 0, 0 \rangle$ N is applied to an object that moves $\vec{d} = \langle 3, 0, 0 \rangle$ m, then the work done is $W = 2(3) + 0(0) + 0(0) = 6$ J
• Simplifies work calculation by accounting for the angle between the force and displacement vectors: $W = |\vec{F}||\vec{d}|\cos\theta$
  • Positive work indicates force and displacement are in the same direction (angle < 90°)
  • Negative work indicates force and displacement are in opposite directions (angle > 90°)
  • Zero work indicates force and displacement are perpendicular (angle = 90°)

Vector properties and operations

• Magnitude of a vector $\vec{a} = \langle a_1, a_2, a_3 \rangle$ is calculated as $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
• Vector components are the individual scalar values that make up a vector (e.g., $a_1$, $a_2$, and $a_3$ for $\vec{a} = \langle a_1, a_2, a_3 \rangle$)
• Scalar multiplication of a vector $\vec{a}$ by a scalar $k$ results in $k\vec{a} = \langle ka_1, ka_2, ka_3 \rangle$, changing the vector's magnitude but not its direction (unless $k < 0$)