Two vectors added using the tip to tail method. There is a 12 meter vector west added to a 5 meter vector north in a right triangle. The hypotenuse is a dashed arrow labeled R. What is the magnitude of the resultant vector? 13 miles 17 miles 60 miles 169 miles
The Correct Answer and Explanation is:
To find the magnitude of the resultant vector R, we are dealing with a right triangle formed by the two given vectors:
- One vector is 12 meters west
- The second vector is 5 meters north
These vectors are perpendicular, meaning they form a right angle when placed tip to tail. The resultant vector R is the hypotenuse of the right triangle.
Step-by-step solution:
Use the Pythagorean theorem:R=(122+52)R = \sqrt{(12^2 + 5^2)}R=(122+52)R=144+25R = \sqrt{144 + 25}R=144+25R=169R = \sqrt{169}R=169R=13R = 13R=13
So, the magnitude of the resultant vector is 13 meters.
Now, look at the answer choices:
- 13 miles
- 17 miles
- 60 miles
- 169 miles
Even though the units in the question are in meters, the answer choices are in miles. Since we are solving for a magnitude, and the triangle is geometric, we can reasonably assume the question is more about the numerical value than the unit. So, 13 miles matches the magnitude of 13 meters.
Explanation
When two vectors are added using the tip-to-tail method, the direction and magnitude of each vector are represented by arrows. To find the resultant vector, you draw the two vectors one after the other, making sure the tip (arrowhead) of the first vector touches the tail (base) of the second vector. Then, draw a new arrow from the starting point of the first vector to the tip of the last vector. This new arrow is called the resultant vector.
In this case, we are given a 12-meter vector pointing west and a 5-meter vector pointing north. Because west and north are at a 90-degree angle to each other, these vectors form a right triangle when placed tip-to-tail. The vector labeled R (the dashed line) is the hypotenuse of this right triangle.
To find the length of the hypotenuse (the magnitude of the resultant vector), we use the Pythagorean theorem, which states that in a right triangle:(Hypotenuse)2=(Side1)2+(Side2)2\text{(Hypotenuse)}^2 = (\text{Side}_1)^2 + (\text{Side}_2)^2(Hypotenuse)2=(Side1)2+(Side2)2
Substituting the values:R=(12)2+(5)2=144+25=169=13R = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13R=(12)2+(5)2=144+25=169=13
Thus, the magnitude of the resultant vector is 13. Therefore, the correct answer is 13 miles.
