Name: adding and multiplying vectors
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

Consider two force vectors in a two-dimensional coordinate system.
These vectors can be added using the parallelogram law, where the tails of the concurrent vectors are joined together and lines parallel to each vector are drawn to form a parallelogram.
The resultant vector can be obtained by drawing a diagonal from the tails to the point of intersection of lines.
Similarly, the vectors can be added using the triangle rule, where two vectors are arranged as two sides of the triangle joined head-to-tail, and the third side of the triangle represents the magnitude and direction of the resultant vector.
Vector subtraction is similar to adding reverse of one vector to another. As a result, the difference between two vectors is obtained using the same rules of vector addition.
When a vector is multiplied by a scalar, the magnitude of the vector varies by the amount of the scalar quantity.
If the scalar is positive, then the direction of the vector remains the same, and if it is negative, the direction is reversed.

vector operations

Vectors are physical quantities that have both magnitude and direction. The vector operations include addition, subtraction, and scalar multiplication.
A vector multiplied by a scalar value is called scalar multiplication. The result obtained is a new vector with a different magnitude. If the scalar is positive, the direction of the vector remains the same, but if it is negative, the direction of the vector is reversed. For example, the product of the mass and velocity yields the momentum. Here, the scalar quantity mass is multiplied by the vector quantity velocity, which results in the vector quantity momentum. The direction of the resultant momentum vector is in the same direction as the velocity vector.
The two vectors can be added geometrically by using the parallelogram law of vector addition. When added using this law, the two vectors are represented as adjacent sides of a parallelogram. The tails of both vectors are placed at the origin, and the lines parallel to each vector are drawn to form a parallelogram. The parallelogram&#39;s diagonal is then drawn from the origin to the opposite corner of the parallelogram to represent the resultant vector. The magnitude and direction of the resultant vector can be determined by measuring the length and angles of the parallelogram&#39;s diagonal. The parallelogram law can be simplified in a special scenario where the two vectors are collinear, meaning they have the same line of action. In this case, the resultant vector can be obtained through an algebraic or scalar addition of the two vectors.
Another method of graphically adding two vectors is the triangle law of vector addition. The triangle rule of vector addition states that if two vectors are arranged joined head to tail as two sides of a triangle, the third side of the triangle represents the magnitude and direction of the resultant vector. Vector addition using this law involves placing the first vector at the origin and placing the second vector head to tail with the first vector. The third side of the triangle from the origin to the end of the second vector represents the resultant vector.
The process of vector subtraction involves adding the reverse (negative) of one vector to another vector. This means that the rules of vector addition can be applied to obtain the difference between the two vectors.

Vector Operations

Vectors are physical quantities that have both magnitude and direction. The vector operations include addition, subtraction, and scalar multiplication.
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Force Vectors

KEY TERMS AND CONCEPTS

adding and multiplying vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses.

A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and velocity is the vector quantity. The result is momentum, and its direction matches the direction of the velocity vector.

Two vectors can be added geometrically with the parallelogram law. In this method, the two vectors are drawn as adjacent sides of a parallelogram with both tails at the origin. Lines are then drawn parallel to each vector to complete the shape.

The diagonal of the parallelogram, drawn from the origin to the opposite corner, shows the resultant vector. Its magnitude and direction can be found by measuring the diagonal's length and angles. When the two vectors are collinear, meaning they share the same line of action, the resultant can be found by algebraic or scalar addition.

Another way to add vectors is the triangle law of vector addition. Here, the vectors are arranged head to tail as two sides of a triangle. The third side of the triangle, from the origin to the end of the second vector, represents the resultant vector.

Vector subtraction uses the reverse, or negative, of one vector added to another vector. This lets the rules of vector addition be used to find the difference between two vectors.

Adding and Multiplying Vectors

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Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses. A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and...

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key terms and concepts

Learn how vector addition, subtraction, and scalar multiplication work using parallelogram and triangle laws, plus a momentum example for high school science.

Adding and Multiplying Vectors | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Adding and Multiplying Vectors

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