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Chapter 3: Problem 31

Given two vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) with magnitudes \(A\) and \(B\), respectively, you can subtract \(\overrightarrow{\mathbf{B}}\) from \(\overrightarrow{\mathbf{A}}\) to get a third vector \(\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\). If the magnitude of \(\overrightarrow{\mathbf{C}}\) is equal to \(C=A+B\), what is the relative orientation of vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathbf{B}}\) ?

Short Answer

Expert verified
Vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are oriented in opposite directions.

Step by step solution

01

Understanding Vector Subtraction

Vector subtraction is performed by reversing the direction of one of the vectors and then performing vector addition. That is, the vector \(\overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{A}} - \overrightarrow{\mathbf{B}}\) can be understood as \(\overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{A}} + (-\overrightarrow{\mathbf{B}})\).
02

Applying the Triangle Inequality

The magnitude of \(\overrightarrow{\mathbf{C}}\), obtained from \(\overrightarrow{\mathbf{A}} - \overrightarrow{\mathbf{B}}\), should satisfy the triangle inequality: \(C \leq A + B.\) In this exercise, we are given that \(C = A + B\).
03

Analyzing Maximum Magnitude Condition

The condition \(C = A + B\) is achieved when the vectors \(\overrightarrow{\mathbf{A}}\) and \(-\overrightarrow{\mathbf{B}}\) are aligned in the same direction. Since \(-\overrightarrow{\mathbf{B}}\) is just \(\overrightarrow{\mathbf{B}}\) reversed, \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) must be oriented in opposite directions to achieve this condition.

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Key Concepts

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

Vector Addition
Vector addition is a fundamental operation in vector mathematics that combines two or more vectors to produce a resultant vector. This operation is akin to path-following where each vector represents a step in a particular direction. To add vectors, consider both their magnitude and direction.

**Constructing the Resultant Vector**
  • Vectors are typically added head-to-tail. Place the tail of the second vector at the head of the first.
  • The resultant vector, or the sum, is then drawn from the tail of the first vector to the head of the last vector in the sequence.
  • If vectors are aligned, the resultant magnitude is simply the sum of individual magnitudes.
Understanding vector addition is crucial, especially in physics and engineering, as it helps resolve forces, velocities, and other vector quantities. It lays the foundation for understanding more complex operations like vector subtraction.
Triangle Inequality
The triangle inequality is an important principle in the context of vectors and geometry. It gives us a rule for the relationship between the lengths of sides in a geometric triangle formed by three vectors. This principle can be summed up as follows:

**Fundamental Rule**
  • If you have three vectors forming a triangle, then the length of any one side must be less than or equal to the sum of the lengths of the other two sides.
  • This idea translates into the statement: \[C \leq A + B\]
In our context, we have \[C = A + B\], indicating that vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are oriented directly opposite one another. This scenario represents a special case where the equality is achieved, meaning no gap between vectors, forming a straight line.
Vector Orientation
Vector orientation describes the positioning or direction a vector points towards. In operations like vector substitution or addition, understanding orientation is key to solving equations effectively.

**Relative Orientation**
  • For maximum resultant magnitude in subtraction, vectors must be in opposite directions.
  • Opposite orientation means the direction of \(\overrightarrow{\mathbf{B}}\) is 180 degrees apart from \(\overrightarrow{\mathbf{A}}\).
When vectors are oriented opposite to each other, subtraction yields a maximized positive magnitude. In geometric terms, visually, this orientation causes the vectors to be structured as a linear path, resulting in the lengths perfectly aligning to create the greatest possible resultant magnitude. This concept helps unlock complex scenarios in physics and engineering by predicting the results of vector interactions.

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Most popular questions from this chapter

A hot air balloon rises vertically with a speed of \(1.5 \mathrm{~m} / \mathrm{s}\). At the same time, there is a horizontal \(10 \mathrm{~km} / \mathrm{h}\) wind blowing. In which direction is the balloon moving?

An astronaut on the Moon fires a projectile from a launcher on a level surface so as to get the maximum range. If the launcher gives the projectile a muzzle velocity of \(25 \mathrm{~m} / \mathrm{s},\) what is the range of the projectile? [Hint: The acceleration due to gravity on the Moon is only onesixth of that on the Earth.]

A 2.05 -m-tall basketball player takes a shot when he is \(6.02 \mathrm{~m}\) from the basket (at the three-point line). If the launch angle is \(25^{\circ}\) and the ball was launched at the level of the player's head, what must be the release speed of the ball for the player to make the shot? The basket is \(3.05 \mathrm{~m}\) above the floor.

A shot-putter launches the shot from a vertical distance of \(2.0 \mathrm{~m}\) off the ground (from just above her ear) at a speed of \(12.0 \mathrm{~m} / \mathrm{s} .\) The initial velocity is at an angle of \(20^{\circ}\) above the horizontal. Assume the ground is flat. (a) Compared to a projectile launched at the same angle and speed at ground level, would the shot be in the air (1) a longer time, (2) a shorter time, or (3) the same amount of time? (b) Justify your answer explicitly; determine the shot's range and velocity just before impact in unit vector (component) notation.

Given two vectors, \(\overrightarrow{\mathrm{A}}\) which has a length of 10.0 and makes an angle of \(45^{\circ}\) below the \(-x\) -axis, and \(\overrightarrow{\mathbf{B}}\) which has an \(x\) -component of +2.0 and \(\mathrm{a} y\) -component of +4.0 (a) sketch the vectors on \(x-y\) axes, with all their "tails" starting at the origin, and (b) calculate \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\).
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