Name: vector projection and the dot product
Uploaded: 2026-06-08T09:39:04.262Z
Duration: 1 min 27 s
Channel: JoVE

The dot product combines two vectors and produces a scalar value.Algebraically, it is calculated by multiplying the corresponding components of the vectors and adding the products. Geometrically, it equals the product of the magnitudes of both vectors and the cosine of the angle between them.A practical application is optimizing the tilt of solar panels for maximum energy absorption. Sunlight is modeled as a three-dimensional direction vector.Another vector represents the panel&#x2019;s normal direction, perpendicular to its surface. Since sunlight travels toward the panel, its vector points inward. To align with the outward normal, the sunlight vector is reversed, and the dot product is calculated between the panel normal and the reversed vector.This means multiplying the corresponding components and adding the results. Dividing this product by the vector magnitudes gives the cosine term that controls energy absorption by measuring how directly sunlight hits the panel.When the angle is small, the cosine is large, meaning greater energy absorption.At 90 degrees, the dot product becomes zero, and the panel receives no direct solar energy.

dot product of two vectors

The dot product, or scalar product, is a fundamental operation in vector algebra that combines two vectors to yield a scalar quantity. It is particularly valuable in physical applications, such as calculating work, and in mathematical contexts, such as determining vector projections and direction cosines.For vectors a = &#x27E8;a&#x2081;, a&#x2082;, a&#x2083;&#x27E9; and b = &#x27E8;b&#x2081;, b&#x2082;, b&#x2083;&#x27E9;, the dot product is defined as:\begin{equation*}\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\end{equation*}This operation is commutative and distributive over vector addition and is compatible with scalar multiplication. If vectors a and b are orthogonal, then their dot product equals zero. When vectors have equal direction, the dot product is maximized for their magnitudes.The dot product also has a geometric definition involving the angle &#x3B8; between two vectors:\begin{equation*}\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\end{equation*}This formulation determines the angle between vectors or checks for orthogonality (&#x3B8; = 90&#xB0; implies a&#xB7;b = 0).The angles between a vector and the positive coordinate axes are known as direction angles. If &#x3B1;, &#x3B2;, and &#x3B3; are the angles between vector a and the x-, y-, and z-axes, respectively, the direction cosines are:\begin{equation*}\cos \alpha = \frac{a_1}{|\mathbf{a}|}, \quad \cos \beta = \frac{a_2}{|\mathbf{a}|}, \quad \cos \gamma = \frac{a_3}{|\mathbf{a}|}\end{equation*}These satisfy the identity:\begin{equation*}\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\end{equation*}The scalar projection of b onto a (comp&#x2090; b) and the vector projection (proj&#x2090; b) extend comp&#x2090; b by scaling the unit vector in the direction of a:\begin{align*}\text{comp}_{\mathbf{a}} \mathbf{b} &amp;= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\\\text{proj}_{\mathbf{a}} \mathbf{b}&amp; = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a}\end{align*}These projections describe how much of b aligns with a, with physical significance in mechanics and electromagnetism.

Dot Product of Two Vectors

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JoVE Core

Calculus

Unit

Vectors in Space

KEY TERMS AND CONCEPTS

vector projection and the dot product

The dot product connects vectors, angles, and projections in vector algebra. It combines two vectors to produce a scalar, which is a single number. This makes it useful in math and in physics, especially for work.

For vectors a = ⟨a1, a2, a3⟩ and b = ⟨b1, b2, b3⟩, the dot product is found by multiplying matching components and adding them: a · b = a1b1 + a2b2 + a3b3. The operation is commutative, so the order does not matter. It is also distributive over vector addition and works with scalar multiplication.

The dot product also has a geometric form. If θ is the angle between two vectors, then a · b = |a||b| cos θ. This form helps find the angle between vectors and check whether they are orthogonal, which means at right angles. When θ = 90°, the dot product is zero. When two vectors point in the same direction, the dot product is largest for their magnitudes.

Direction angles describe the angles between a vector and the positive x-, y-, and z-axes. If α, β, and γ are those angles for vector a, then the direction cosines are cos α = a1/|a|, cos β = a2/|a|, and cos γ = a3/|a|. These values satisfy cos²α + cos²β + cos²γ = 1.

The dot product also leads to projection. The scalar projection, or compa b, tells how much of b lies along a, and it is given by compa b = (a · b)/|a|. The vector projection, or proja b, turns that amount into a vector by scaling the unit vector in the direction of a: proja b = (a · b)/|a|² a. These projections show how strongly one vector aligns with another and are important in mechanics and electromagnetism.

Vector Projection and the Dot Product

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The dot product connects vectors, angles, and projections in vector algebra. It combines two vectors to produce a scalar, which is a single number. This makes it useful in math and in physics, especially for work. For vectors a = ⟨a1, a2, a3⟩ and b = ⟨b1, b2, b3⟩, the dot product is found by multiplying matching components and adding them: a · b = a1b1 + a2b2 + a3b3. The operation is commutative, so the order does not matter. It is also distributive over vector addition and works with scalar...

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key terms and concepts

Learn how the dot product links vectors, angles, direction cosines, and projections, with clear formulas and key uses in physics and math.

Vector Projection and the Dot Product | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Vector Projection and the Dot Product

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