Glossary

14.2 Dot product and vector projections

3 min read•july 22, 2024

The 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

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  • Computes a scalar value by multiplying corresponding components of two vectors a⃗\vec{a} and b⃗\vec{b} and summing the results
    • In 2D: a⃗⋅b⃗=axbx+ayby\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y (horizontal and vertical components)
    • In 3D: a⃗⋅b⃗=axbx+ayby+azbz\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
    • a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta where θ\theta is the angle between a⃗\vec{a} and b⃗\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 θ\theta between two vectors: cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}
    • Use the inverse cosine (arccos) to isolate θ\theta: θ=arccos⁡(a⃗⋅b⃗∣a⃗∣∣b⃗∣)\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: a⃗=⟨1,0⟩\vec{a} = \langle 1, 0 \rangle and b⃗=⟨0,1⟩\vec{b} = \langle 0, 1 \rangle are orthogonal since a⃗⋅b⃗=1(0)+0(1)=0\vec{a} \cdot \vec{b} = 1(0) + 0(1) = 0

Vector Projections

Vector projection and significance

  • The projection of a⃗\vec{a} onto b⃗\vec{b}, written as projb⃗a⃗\text{proj}_{\vec{b}}\vec{a}, is a vector parallel to b⃗\vec{b} with magnitude equal to the component of a⃗\vec{a} in the direction of b⃗\vec{b}
    • Calculated using projb⃗a⃗=a⃗⋅b⃗∣b⃗∣2b⃗\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 of a⃗\vec{a} onto b⃗\vec{b} is the signed magnitude of the vector projection
    • Calculated using compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}
    • Positive scalar projection means a⃗\vec{a} has a component in the same direction as b⃗\vec{b}
    • Negative scalar projection means a⃗\vec{a} has a component in the opposite direction of b⃗\vec{b}

Applications in geometric problem-solving

  1. Calculate F⃗\vec{F} over a displacement d⃗\vec{d} using W=F⃗⋅d⃗W = \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 F⃗\vec{F} in a specific direction d⃗\vec{d} using F⃗component=projd⃗F⃗\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

Key Terms to Review (17)

|v|: |v| represents the magnitude or length of a vector v. This value is crucial in understanding the size and direction of vectors, allowing us to calculate their properties in various mathematical contexts, including dot products and projections. The magnitude is computed using the square root of the sum of the squares of its components, which plays a vital role in many applications involving vectors.
Angle between vectors: The angle between vectors is the measure of the smallest angle formed when two vectors are placed tail to tail. This angle is significant in understanding the relationship between the vectors, as it indicates how closely aligned they are with each other. The concept of the angle between vectors is closely related to operations involving vectors, particularly in calculating the dot product and vector projections, which help in determining not only the magnitude but also the directional relationships between them.
Cauchy-Schwarz Inequality: The Cauchy-Schwarz Inequality states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\), the absolute value of their dot product is less than or equal to the product of their magnitudes. Mathematically, it can be expressed as \( |\mathbf{u} \cdot \mathbf{v}| \leq ||\mathbf{u}|| \, ||\mathbf{v}|| \). This fundamental inequality is crucial in understanding geometric relationships and projections in vector spaces.
Commutative Property: The commutative property states that the order in which two numbers are added or multiplied does not change the sum or product. This concept is fundamental in mathematics and is utilized in various operations, allowing for flexibility in problem-solving and simplifying expressions.
Cosine of the angle: The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This concept connects directly to various applications in geometry, particularly in understanding relationships between angles and sides, as well as in computations involving dot products and vector projections.
Distributive Property: The distributive property is a fundamental algebraic principle that states that multiplying a number by a sum or difference can be achieved by distributing the multiplication across each term within the parentheses. This property is vital in simplifying expressions and solving equations, playing a significant role in various mathematical contexts including geometry, vectors, and dot products.
Dot product: The dot product is a mathematical operation that takes two vectors and returns a scalar, which represents the product of their magnitudes and the cosine of the angle between them. This operation helps in understanding how vectors relate to each other in terms of direction and length, making it essential for various applications like projections and determining orthogonality.
Formula for projection: The formula for projection refers to a mathematical expression used to find the projection of one vector onto another. This concept helps in determining how much of one vector lies in the direction of another, revealing important geometric relationships between vectors. The formula is crucial in various applications, including physics and computer graphics, where understanding directional components is essential.
Length of projection: The length of projection refers to the scalar value that represents the magnitude of one vector along the direction of another vector. It is calculated using the dot product and provides insight into how much of one vector lies in the direction of another, making it crucial for understanding vector relationships and spatial dimensions.
Orthogonal projection: Orthogonal projection is the process of projecting a vector onto a subspace in such a way that the resulting vector is as close as possible to the original vector while being perpendicular to the subspace. This concept is closely tied to the idea of minimizing the distance between the original vector and its projection, which can be calculated using the dot product. The resulting projection maintains key properties of vector relationships and helps in understanding geometric interpretations in higher dimensions.
Projection of a Vector: The projection of a vector onto another vector is a way to express how much of one vector goes in the direction of another. It involves breaking down the first vector into two components: one that is parallel to the second vector and another that is perpendicular to it. This concept is crucial in understanding relationships between vectors and is often calculated using the dot product.
Scalar projection: Scalar projection is the measure of the length of the shadow a vector projects onto another vector, representing how much one vector extends in the direction of another. This concept is important in understanding relationships between vectors, particularly in applications like physics and engineering where direction and magnitude are key. It is calculated using the dot product, which quantifies how two vectors align with each other.
Û: In the context of vector mathematics, $$\hat{u}$$ represents a unit vector, which is a vector that has been normalized to have a magnitude of 1. Unit vectors are crucial in various applications, particularly in defining direction without concern for the size of the vector. They simplify calculations involving vector projections and help in visualizing directions in space.
Unit Vector: A unit vector is a vector that has a magnitude of exactly one. It is often used to indicate direction without concern for length, making it essential in various applications, such as normalizing vectors or expressing directional components in geometry. By converting any vector into a unit vector, we can simplify calculations involving angles and projections, particularly when dealing with dot products and other vector operations.
V · w: The term v · w represents the dot product of two vectors v and w. This operation results in a scalar value that quantifies the extent to which the two vectors align with each other, providing insight into their directional relationship and magnitude. The dot product is calculated as the sum of the products of the corresponding components of the vectors and has applications in various fields including physics, computer graphics, and engineering.
Vector magnitude: Vector magnitude is a measure of the length or size of a vector, representing the distance from the origin to the point defined by the vector in a coordinate system. This concept is crucial for understanding how vectors interact with each other, particularly in operations like the dot product and when projecting one vector onto another.
Work done by a force: Work done by a force is the measure of energy transfer that occurs when a force is applied to an object and the object moves in the direction of the force. It can be calculated using the formula $$W = F imes d imes ext{cos}( heta)$$, where W is work, F is the magnitude of the force, d is the distance moved by the object, and $$ heta$$ is the angle between the direction of the force and the direction of motion. Understanding work in this way connects to concepts such as energy transfer and vector projections, as it involves both magnitude and direction.
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