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Chapter 16: Problem 5

Find a single vector resulting from the operations. 3(5,7,13)

Short Answer

Expert verified
Answer: The resulting vector is (15, 21, 39).

Step by step solution

01

Multiply the vector by the scalar

Multiply each component of the vector (5,7,13) by the scalar 3.
02

Calculate the resulting vector

After multiplying each component of the vector (5,7,13) by 3, we get: - 3 * 5 = 15 - 3 * 7 = 21 - 3 * 13 = 39 So, the resulting vector is (15, 21, 39).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
Scalar multiplication is an operation that involves multiplying a vector by a scalar, which is essentially just a single number. This process scales each component of the vector by the same scalar, effectively stretching or shrinking the vector without altering its direction.
For instance, if you have a vector \((a, b, c)\) and you multiply it by a scalar \(k\), each component of the vector is multiplied individually to give you \((ka, kb, kc)\).
  • This means that a vector such as \((5, 7, 13)\) when multiplied by the scalar \(3)\) results in \((3 \times 5, 3 \times 7, 3 \times 13)\).
  • The resulting vector will have components that are each three times the value of the original components.
The entire shape of the vector remains similar in terms of direction, but with its magnitude scaled up by a factor of \(3\).
Vector Components
Vectors are fundamental objects used in mathematics and physics to represent quantities having both magnitude and direction. Each vector contains components, which are essentially the coordinates that describe the vector's position or direction along each axis.
For example, a vector \((5, 7, 13)\) has three components corresponding to its position along the x, y, and z axes. Each of these numbers tells us how far in each direction the vector points from the origin.
  • The component \(5\) represents the x-axis distance, \(7\) represents the y-axis distance, and \(13\) represents the z-axis distance.
  • When you perform operations like scalar multiplication, you modify these components directly, altering the vector's magnitude without changing its inherent direction.
Understanding vector components is essential for performing precise calculations, such as in physics, where they help to describe forces and motions in space.
Resulting Vector
After performing vector operations such as scalar multiplication, you end up with what is known as the resulting vector. This new vector demonstrates the effects of the operations carried out on the original vector.
In our example, multiplying the vector \((5, 7, 13)\) by a scalar of \(3\) yields a resulting vector of \((15, 21, 39)\).
  • Each element of the original vector is scaled by the scalar, resulting in components that are three times larger.
  • The operation leaves the direction unchanged but increases the size of the vector by the scalar factor.
This resulting vector is essential in various fields, including physics and engineering, where such calculations are necessary to model real-world phenomena like forces acting on an object or the velocity of a moving body. The ability to compute a resulting vector efficiently and accurately is a crucial skill in the study of vectors.

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Most popular questions from this chapter

Find a single vector resulting from the operations. \(2 \cdot 3(5,7,13)\)

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