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Chapter 1: Problem 85

Find the magnitude and direction of (a) \(9 \vec{B}-3 \vec{A}\) and (b) \(-5 \vec{A}+8 \vec{B},\) where \(\vec{A}=(23.0,59.0), \vec{B}=(90.0,-150.0)\)

Short Answer

Expert verified
Answer: The magnitude of the vector \(9\vec{B} - 3\vec{A}\) is approximately \(1664.43\), and its direction is approximately \(69.76^\circ\) clockwise from the positive x-axis. The magnitude of the vector \(-5\vec{A} + 8\vec{B}\) is approximately \(1370.98\), and its direction is approximately \(68.05^\circ\) clockwise from the positive x-axis.

Step by step solution

01

Calculate the new vector

We need to find the vector resulting from the expression \(9\vec{B} - 3\vec{A}\). To do this, we will multiply each vector by the corresponding scalar and then subtract them.
02

Determine the components of the new vector

Calculate the components of the new vector by performing the operations: \((9 * 90 - 3 * 23, 9 * -150 - 3 * 59)\). This results in the new vector \(\vec{C} = (561, -1539)\).
03

Calculate the magnitude of the new vector

The magnitude of a vector can be computed using the Pythagorean theorem applied to its components: \(|\vec{C}| = \sqrt{C_x^2 + C_y^2}\). So, \(|\vec{C}| = \sqrt{561^2 + (-1539)^2} \approx 1664.43\).
04

Calculate the angle needed to determine the direction

We will use the arctangent function to find the angle: \(\theta = \arctan{\frac{C_y}{C_x}} = \arctan{\frac{-1539}{561}}\).
05

Find the direction of the vector

Calculate the direction: \(\theta = \arctan{\frac{-1539}{561}} \approx -69.76^\circ\). Since the angle is negative, the direction is clockwise from the positive x-axis.
06

Present the final answer

The magnitude of the vector \(9\vec{B} - 3\vec{A}\) is approximately \(1664.43\). The direction of this vector is approximately \(69.76^\circ\) clockwise from the positive x-axis. #b)#
07

Calculate the new vector

We need to find the vector resulting from the expression \(-5\vec{A} + 8\vec{B}\). To do this, we will multiply each vector by the corresponding scalar and then add them.
08

Determine the components of the new vector

Calculate the components of the new vector by performing the operations: \((-5 * 23 + 8 * 90, -5 * 59 + 8 * -150)\). This results in the new vector \(\vec{D} = (515, -1269)\).
09

Calculate the magnitude of the new vector

The magnitude of a vector can be computed using the Pythagorean theorem applied to its components: \(|\vec{D}| = \sqrt{D_x^2 + D_y^2}\). So, \(|\vec{D}| = \sqrt{515^2 + (-1269)^2} \approx 1370.98\).
10

Calculate the angle needed to determine the direction

We will use the arctangent function to find the angle: \(\theta = \arctan{\frac{D_y}{D_x}} = \arctan{\frac{-1269}{515}}\).
11

Find the direction of the vector

Calculate the direction: \(\theta = \arctan{\frac{-1269}{515}} \approx -68.05^\circ\). Since the angle is negative, the direction is clockwise from the positive x-axis.
12

Present the final answer

The magnitude of the vector \(-5\vec{A} + 8\vec{B}\) is approximately \(1370.98\). The direction of this vector is approximately \(68.05^\circ\) clockwise from the positive x-axis.

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