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Chapter 1: Problem 27

Determine the real number \(\alpha\) such that vectors \(\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{b}=9 \mathbf{i}+\alpha \mathbf{j}\) are orthogonal.

Short Answer

Expert verified
The real number \(\alpha\) such that the vectors are orthogonal is \(-6\).

Step by step solution

01

Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. This means that we will find the dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) and set it to zero to solve for \(\alpha\).
02

Dot Product of Vectors

The dot product of two vectors \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b} = 9\mathbf{i} + \alpha \mathbf{j}\) is calculated as follows:\[\mathbf{a} \cdot \mathbf{b} = (2)(9) + (3)(\alpha) = 18 + 3\alpha\].
03

Setting the Dot Product to Zero

Since the vectors are orthogonal, we set their dot product to zero:\[ 18 + 3\alpha = 0 \].
04

Solve for \(\alpha\)

Solving the equation \( 18 + 3\alpha = 0 \) for \(\alpha\):1. Subtract 18 from both sides:\[ 3\alpha = -18 \].2. Divide both sides by 3:\[ \alpha = -6 \].

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Key Concepts

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

Understanding the Dot Product
The dot product is a fundamental operation in vector algebra. It helps us to find whether two vectors are orthogonal or not. To compute the dot product of two vectors, you multiply their corresponding components and then sum those products. If we have vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is given by:\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2\]Key aspects of the dot product include:
  • If the dot product is zero, the vectors are orthogonal or perpendicular to each other.
  • A non-zero dot product indicates the vectors are not orthogonal.
Orthogonal vectors are at right angles to each other, which is essential in various applications, including finding projections and determining angles.
Exploring Vector Algebra
Vector algebra is a powerful tool used to analyze magnitudes and directions. It involves operations such as addition, subtraction, and the dot product of vectors, among others.When working with vectors:
  • Addition: Vectors are added by summing their corresponding components. For example, \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) gives \( \mathbf{a} + \mathbf{b} = (a_1+b_1) \mathbf{i} + (a_2+b_2) \mathbf{j} \).
  • Subtraction: Vector subtraction works similarly to addition, except we subtract the components.
  • Magnitude: The magnitude of a vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) is calculated as \( \sqrt{a_1^2 + a_2^2} \).
Understanding these operations can help in various physics and engineering applications, allowing you to solve complex problems involving force, velocity, and other vectors.
Solving Linear Equations in Vector Context
In vector problems, especially those involving orthogonality, solving linear equations is crucial to find unknowns. Consider the scenario in which you need to determine a constant, like \( \alpha \) in our original problem.Here's how to approach solving linear equations when applied to vector cases:
  • Identify the Equation: Determine which equation needs to be solved. For orthogonal vectors, the dot product should be set to zero as vectors are perpendicular.
  • Isolate the Variable: Once the equation is set up, bring terms involving the variable to one side of the equation. Move constants to the opposite side.
  • Solve the Equation: Perform algebraic manipulations such as addition, subtraction, multiplication, or division to isolate and solve for the unknown. For example, \( 18 + 3\alpha = 0 \) implies solving for \( \alpha \).
Solving such equations helps find specific values, which play critical roles in practical applications, like determining forces or coordinates in physics.

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Most popular questions from this chapter

The ring torus symmetric about the \(z\) -axis is a special type of surface in topology and its equation is given by \(\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4 R^{2}\left(x^{2}+y^{2}\right)\), where \(R>r>0 .\) The numbers \(R\) and \(r\) are called are the major and minor radii, respectively, of the surface. The following figure shows a ring torus for which \(R=2\) and \(r=1\). a. Write the equation of the ring torus with \(R=2\) and \(r=1\), and use a CAS to graph the surface. Compare the graph with the figure given. b. Determine the equation and sketch the trace of the ring torus from a. on the \(x y\) -plane. c. Give two examples of objects with ring torus shapes.

Rewrite the given equation of the quadric surface in standard form. Identify the surface. $$ 5 x^{2}-4 y^{2}+20 z^{2}=0 $$

An airplane is flying in the direction of \(43^{\circ}\) east of north (also abbreviated as N43E) at a speed of 550 mph. A wind with speed 25 mph comes from the southwest at a bearing of N15E. What are the ground speed and new direction of the airplane?

Consider \(\mathbf{r}(t)=\langle\cos t, \sin t, 2 t\rangle\) the position vector of a particle at time \(t \in[0,30]\), where the components of are expressed in centimeters and time in seconds. Let \(\overrightarrow{O P}\) be the position vector of the particle after \(1 \mathrm{sec}\). a. Show that all vectors \(\overrightarrow{P Q}\), where \(Q(x, y, z)\) is an arbitrary point, orthogonal to the instantaneous velocity vector \(\mathbf{v}(1)\) of the particle after \(1 \mathrm{sec}\), can be expressed as \(\overrightarrow{P Q}=\langle x-\cos 1, y-\sin 1, z-2\rangle\), where \(x \sin 1-y \cos 1-2 z+4=0 .\) The set of point \(Q\) describes a plane called the normal plane to the path of the particle at point \(P\) b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point \(P\) along with the path of the particle.

The intersection between cylinder \((x-1)^{2}+y^{2}=1\) and sphere \(x^{2}+y^{2}+z^{2}=4\) is called a Viviani curve. a. Solve the system consisting of the equations of the surfaces to find the equation of the intersection curve. (Hint: Find \(x\) and \(y\) in terms of \(z .)\) b. Use a computer algebra system (CAS) to visualize the intersection curve on sphere \(x^{2}+y^{2}+z^{2}=4\).
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