Let $| u |=4$ at an angle of $210^{\circ}$ and $| v |=9$ at an angle of $315^{\circ}$, and $w = u - v$. What is the magnitude and direction angle of $w$?
A. $|w|=5.5 ; \theta=156.1^{\circ}$
B. $|w|=5.5 ; \theta=203.9^{\circ}$
C. $|w|=10.8 ; \theta=156.1^{\circ}$
D. $|w|=10.8 ; \theta=203.9^{\circ}$
Answer (1)
Convert vectors u and v from polar to rectangular coordinates using trigonometric functions.
Calculate vector w by subtracting vector v from vector u in rectangular coordinates.
Compute the magnitude of vector w using the Pythagorean theorem.
Determine the direction angle of vector w using the arctangent function, adjusting for the correct quadrant: ∣ w ∣ = 10.8 ; θ = 156. 1 ∘ .
Explanation
Problem Setup We are given two vectors, u and v , in polar form. We have ∣ u ∣ = 4 at an angle of 21 0 ∘ and ∣ v ∣ = 9 at an angle of 31 5 ∘ . We want to find the magnitude and direction angle of w = u − v .
Convert to Rectangular Coordinates First, we convert the vectors u and v from polar to rectangular coordinates. Recall that for a vector with magnitude r and angle θ , the rectangular coordinates are given by ( r cos ( θ ) , r sin ( θ )) .
For u , we have: u x = ∣ u ∣ cos ( 21 0 ∘ ) = 4 cos ( 21 0 ∘ ) = 4 ⋅ ( − 2 3 ) = − 2 3 u y = ∣ u ∣ sin ( 21 0 ∘ ) = 4 sin ( 21 0 ∘ ) = 4 ⋅ ( − 2 1 ) = − 2 So, u = ( − 2 3 , − 2 ) .
For v , we have: v x = ∣ v ∣ cos ( 31 5 ∘ ) = 9 cos ( 31 5 ∘ ) = 9 ⋅ 2 2 = 2 9 2 v y = ∣ v ∣ sin ( 31 5 ∘ ) = 9 sin ( 31 5 ∘ ) = 9 ⋅ ( − 2 2 ) = − 2 9 2 So, v = ( 2 9 2 , − 2 9 2 ) .
Calculate w = u - v Next, we find the rectangular coordinates of w = u − v :
w x = u x − v x = − 2 3 − 2 9 2 ≈ − 3.464 − 6.364 = − 9.828 w y = u y − v y = − 2 − ( − 2 9 2 ) = − 2 + 2 9 2 ≈ − 2 + 6.364 = 4.364 So, w = ( − 2 3 − 2 9 2 , − 2 + 2 9 2 ) .
Calculate Magnitude of w Now, we calculate the magnitude of w :
∣ w ∣ = w x 2 + w y 2 = ( − 2 3 − 2 9 2 ) 2 + ( − 2 + 2 9 2 ) 2 ≈ ( − 9.828 ) 2 + ( 4.364 ) 2 = 96.59 + 19.04 = 115.63 ≈ 10.75 Therefore, ∣ w ∣ ≈ 10.75 .
Calculate Direction Angle of w Finally, we calculate the direction angle θ of w :
θ = arctan ( w x w y ) = arctan ( − 2 3 − 2 9 2 − 2 + 2 9 2 ) ≈ arctan ( − 9.828 4.364 ) ≈ − 23.9 4 ∘ Since w x < 0 and 0"> w y > 0 , w is in the second quadrant. Therefore, we add 18 0 ∘ to the angle: θ = − 23.9 4 ∘ + 18 0 ∘ = 156.0 6 ∘ Therefore, the direction angle of w is approximately 156.0 6 ∘ .
Final Answer The magnitude of w is approximately 10.75 and the direction angle is approximately 156.0 6 ∘ .
Examples
Understanding vector subtraction is crucial in physics, especially when analyzing forces acting on an object. For instance, imagine a kite being pulled by the wind (vector u ) and held by a person (vector v ). The resultant force ( w = u − v ) determines the kite's stability and direction. By calculating the magnitude and direction of w , we can predict how the kite will behave under these combined forces, ensuring it stays aloft and doesn't drift uncontrollably.
