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Scalar Product (Dot Product)

Scalar Product (Dot Product)

The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a number, that is, a scalar quantity. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector.
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The dot product is an essential concept in mathematics and physics.
In engineering, the dot product of any two vectors is the product of the magnitudes of the vectors and the cosine of the angle between them. It is denoted by a dot symbol between the two vectors.
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Related Experiment Video

Updated: Jan 23, 2026

Construction of Vapor Chambers Used to Expose Mice to Alcohol During the Equivalent of all Three Trimesters of Human Development

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Measurements in First-Trimester Abortion Products: A Pathologic Study.

Rosete Nogueira¹, Sara Sousa¹, Ana Cristina Braga¹

¹From the Surgical Sciences Domain, Medicine School (Drs Nogueira and Pinto) and the Engineering School, Department of Production and Systems (Dr Braga), Minho University, Campus de Gualtar, Braga, Portugal; the Life and Health Sciences Research Institute (ICVS), ICVS/3B's-PT Government Associate Laboratory, Braga/Guimarães, Portugal (Drs Nogueira and Pinto); the Embryo-fetal Pathology Laboratory (Dr Nogueira and Mss Sousa and Azevedo) and the Prenatal Clinical Department (Dr Tavares), CGC Genetics, Porto, Portugal; the Prenatal Diagnosis Department, Hospital S Teotónio, Viseu, Portugal (Dr Pereira); the Prenatal Diagnosis Department, Hospital Tâmega Sousa, Penafiel, Portugal (Dr Carmo); and the Department of Pediatric Surgery, Hospital de S. Marcos, Braga, Portugal (Dr Pinto).

Archives of Pathology & Laboratory Medicine

|June 8, 2019

Summary

This summary is machine-generated.

This study establishes normative measurements for first-trimester spontaneous abortions, providing crucial data for evaluating early pregnancy loss and validating prenatal diagnostic tools.

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Area of Science:

Pathology
Embryology
Prenatal Diagnosis

Background:

Advances in prenatal diagnosis necessitate better understanding of early pregnancy.
Medically challenging situations in early pregnancy require accurate pathologic assessment.
First-trimester abortion products are increasingly studied.

Purpose of the Study:

To evaluate measurements of first-trimester spontaneous abortion specimens.
To validate embryo and gestational-sac measurements against gestational age (GA).
To establish normative data for pathologic assessment.

Main Methods:

Retrospective case study of 585 gestational sacs, 182 embryos, and 116 umbilical cords.
Specimens collected between June 2015 and April 2017.
Regression techniques used to model measurements including weight, diameter, and crown-rump length.

Main Results:

Normative percentiles for gestational-sac diameter and umbilical cord length were calculated by GA.
Average increases: 3 mm/week for sac diameter, 1.35 mm/week for cord length.
Embryo crown-rump length measurements provided for GA 6-12 weeks.

Conclusions:

Pathologic measurements should be compared with expected values.
Correlation with ultrasound, clinical, and microscopic findings is essential.
Deviations from expected measurements aid in understanding early pregnancy loss.