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Chapter 12: Problem 7

Explain how to find a scalar multiple of a vector geometrically.

Short Answer

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Question: Explain how to find a scalar multiple of a vector geometrically using the given step by step solution. Answer: To find a scalar multiple of a vector geometrically, we first need to understand that the scalar multiple is obtained by resizing the vector without changing its direction. Follow these steps: 1. Draw the original vector on a Cartesian coordinate system, starting at the origin and pointing to its coordinates. 2. Choose a scalar factor to multiply the original vector by. 3. Multiply each component of the original vector by the scalar factor to find the scalar multiple's coordinates. 4. Draw the scalar multiple vector on the same coordinate system, starting at the origin and pointing to the new coordinates. 5. Observe the relationship between the original vector and the scalar multiple vector, noting their proportional lengths and similar direction.

Step by step solution

01

Understanding Scalar Multiples

A scalar multiple of a vector is the result of multiplying a vector by a scalar (a real number). This operation will resize the vector, either stretching or compressing its length, without changing its direction.
02

Draw the Original Vector

Choose a vector to work with. For example, let's use the vector v = (3,2). Draw this vector on a Cartesian coordinate system using a straight line with an arrowhead. The tail of the vector should start at the origin (0,0), and the arrowhead should point to the coordinates (3,2).
03

Choose the Scalar Factor

Decide on the scalar value that you will multiply the original vector with. For example, let's use the scalar factor k = 2. This means we want to find the vector 2v or, in other words, stretch the length of the original vector by a factor of 2 without changing its direction.
04

Compute Scalar Multiple Coordinates

To find the coordinates of the scalar multiple of the vector, multiply each component of the original vector by the scalar factor. Using our example, 2v = (2 * 3, 2 * 2) = (6,4).
05

Draw the Scalar Multiple Vector

Now draw the scalar multiple vector on the same Cartesian coordinate system. With the tail starting at the origin (0,0), the arrowhead should point to the coordinates (6,4). This new vector represents the scalar multiple of the original vector and should be in the same direction but twice as long.
06

Observe the Relationship and Similarity

Observe the resulting scalar multiple vector and its relationship to the original vector. They should have the same direction and the lengths should be proportional to the scalar factor, k = 2 in our case. Compare the original vector to the scalar multiple, noting their geometric similarity. By following these steps, we have found the scalar multiple of a vector geometrically.

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