The dot product is a powerful tool for analyzing vector relationships. It calculates a scalar value by multiplying and summing vector components, revealing how vectors align. This operation helps determine angles between vectors and their relative orientations.

Vector projections build on the dot product, allowing us to break down vectors into useful components. By projecting one vector onto another, we can solve real-world problems involving forces, distances, and motion on inclined planes.

Dot Product

Dot product calculation and meaning

• Computes a scalar value by multiplying corresponding components of two vectors $\vec{a}$ and $\vec{b}$ and summing the results
  • In 2D: $\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y$ (horizontal and vertical components)
  • In 3D: $\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z$ (includes depth component)
• Represents the projection of one vector onto another scaled by the magnitude of the other vector
  • $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
  • Positive dot product indicates vectors point in similar directions (acute angle)
  • Negative dot product indicates vectors point in opposite directions (obtuse angle)

Angle determination using dot product

• Rearrange the projection formula to solve for the angle $\theta$ between two vectors: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
  • Use the inverse cosine (arccos) to isolate $\theta$: $\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$
• Dot product of zero means vectors are orthogonal (perpendicular) with a 90° angle between them
  • Example: $\vec{a} = \langle 1, 0 \rangle$ and $\vec{b} = \langle 0, 1 \rangle$ are orthogonal since $\vec{a} \cdot \vec{b} = 1(0) + 0(1) = 0$

Vector Projections

Vector projection and significance

• The projection of $\vec{a}$ onto $\vec{b}$, written as $\text{proj}_{\vec{b}}\vec{a}$, is a vector parallel to $\vec{b}$ with magnitude equal to the component of $\vec{a}$ in the direction of $\vec{b}$
  • Calculated using $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$
  • Decomposes a vector into components parallel and perpendicular to another vector
• The scalar projection of $\vec{a}$ onto $\vec{b}$ is the signed magnitude of the vector projection
  • Calculated using $\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
  • Positive scalar projection means $\vec{a}$ has a component in the same direction as $\vec{b}$
  • Negative scalar projection means $\vec{a}$ has a component in the opposite direction of $\vec{b}$

Applications in geometric problem-solving

1. Calculate work done by a force $\vec{F}$ over a displacement $\vec{d}$ using $W = \vec{F} \cdot \vec{d}$

   • Only the component of the force in the direction of motion contributes to work

2. Find the component of a force $\vec{F}$ in a specific direction $\vec{d}$ using $\vec{F}{\text{component}} = \text{proj}{\vec{d}}\vec{F}$

   • Useful for resolving forces on inclined planes into parallel and perpendicular components

3. Determine the distance between a point and a line or plane by projecting the point onto the normal vector of the line or plane

   • The scalar projection gives the signed distance (positive for one side, negative for the other)

4. Analyze the forces acting on an object resting on an inclined plane

   • The projection of the object's weight onto the plane is the force component parallel to the surface
   • The perpendicular component of the weight is balanced by the normal force from the plane