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Chapter 12: Problem 6

Explain how to add two vectors geometrically.

Short Answer

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Question: Explain how to add two vectors geometrically using either the parallelogram rule or the triangle rule. Answer: To add two vectors geometrically, you can either use the parallelogram rule or the triangle rule. For the parallelogram rule, represent both vectors as adjacent sides of a parallelogram and draw a diagonal from their common starting point to the opposite corner; that diagonal represents the sum of the two vectors. For the triangle rule, draw the second vector with its tail at the head of the first vector, making them consecutive sides of a triangle, and then draw the third side connecting the tail of the first vector to the head of the second; this third side represents the sum of the two vectors. In both cases, measure the length and direction of the resulting vector to find the magnitude and direction of the sum.

Step by step solution

01

Understand the concept of vectors geometrically

A vector is a quantity that has both magnitude (length) and direction. Geometrically, a vector can be represented as an arrow with a specified length and direction.
02

Using the Parallelogram Rule

The parallelogram rule for adding vectors states that if two vectors are represented as the sides of a parallelogram, the sum of the vectors is given by the diagonal of the parallelogram. 1. Choose a suitable scale to represent the magnitude of the vectors. 2. Draw the first vector as an arrow with the length representing its magnitude and pointing in the specified direction. 3. Draw the second vector also as an arrow with the length representing its magnitude and pointing in the specified direction. 4. Use the two vectors as adjacent sides of a parallelogram. Complete the parallelogram by drawing parallel lines to their respective vectors. 5. Draw a diagonal from the common vertex where both vectors start to the opposite vertex of the parallelogram. 6. The diagonal represents the sum of the vectors, both in magnitude and direction. Measure its length and direction using the same scale chosen in the first step.
03

Using the Triangle Rule

The triangle rule for adding vectors states that if two vectors are represented as the sides of a triangle (tail of the second vector placed at the head of the first vector), the sum of the vectors is given by the third side of the triangle. 1. Choose a suitable scale to represent the magnitude of the vectors. 2. Draw the first vector as an arrow with the length representing its magnitude and pointing in the specified direction. 3. From the head (arrow tip) of the first vector, draw the second vector also as an arrow with the length representing its magnitude and pointing in the specified direction. 4. Now, draw the third side of the triangle, which starts at the tail of the first vector and ends at the head of the second vector. 5. The third side represents the sum of the vectors, both in magnitude and direction. Measure its length and direction using the same scale chosen in the first step. By following either the parallelogram rule or the triangle rule, we can geometrically add two vectors.

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Most popular questions from this chapter

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. What conditions on \(\mathbf{u}\) and \(\mathbf{v}\) lead to equality in the Cauchy-Schwarz Inequality?

Alternative derivation of the curvature Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that \(\left.\mathbf{v} \times \mathbf{a}=\kappa|\mathbf{v}|^{3} \mathbf{B} . \text { (Note that } \mathbf{T} \times \mathbf{T}=\mathbf{0} .\right)\) b. Solve the equation in part (a) for \(\kappa\) and conclude that \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{\left|\mathbf{v}^{3}\right|},\) as shown in the text.

An object moves along an ellipse given by the function \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(a > 0\) and \(b > 0\) a. Find the velocity and speed of the object in terms of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) b. With \(a=1\) and \(b=6,\) graph the speed function, for \(0 \leq t \leq 2 \pi .\) Mark the points on the trajectory at which the speed is a minimum and a maximum. c. Is it true that the object speeds up along the flattest (straightest) parts of the trajectory and slows down where the curves are sharpest? d. For general \(a\) and \(b\), find the ratio of the maximum speed to the minimum speed on the ellipse (in terms of \(a\) and \(b\) ).

\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Express \(\mathbf{I}\) and \(\mathbf{J}\) in terms of the usual unit coordinate vectors i and j. Then, write i and \(\mathbf{j}\) in terms of \(\mathbf{I}\) and \(\mathbf{J}\).

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle\)
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