Given that [tex]\overrightarrow{O P}=\binom{4}{3}[/tex] and the point [tex]Q[/tex] is [tex](16,8)[/tex] find [tex]|\overrightarrow{P Q}|[/tex]
Answer (2)
Calculate the coordinates of point Q: ( 16 , 8 ) 2 = ( 256 , 64 ) .
Find the vector PQ by subtracting OP from OQ : PQ = ( 64 256 ) − ( 3 4 ) = ( 61 252 ) .
Calculate the magnitude of the vector PQ : ∣ PQ ∣ = ( 252 ) 2 + ( 61 ) 2 = 67225 .
Simplify to find the final answer: ∣ PQ ∣ ≈ 259.28 .
259.28
Explanation
Problem Analysis and Setup We are given the vector OP = ( 3 4 ) and the point Q with coordinates ( 16 , 8 ) 2 . We need to find the magnitude of the vector PQ . First, we need to calculate the coordinates of point Q .
Calculating Coordinates of Q The coordinates of point Q are given by ( 16 , 8 ) 2 = ( 1 6 2 , 8 2 ) = ( 256 , 64 ) . Therefore, the vector OQ is given by OQ = ( 64 256 ) .
Finding Vector PQ Now we can find the vector PQ by subtracting OP from OQ : PQ = OQ − OP = ( 64 256 ) − ( 3 4 ) = ( 64 − 3 256 − 4 ) = ( 61 252 )
Calculating Magnitude of PQ Next, we calculate the magnitude of the vector PQ using the formula: ∣ PQ ∣ = ( 252 ) 2 + ( 61 ) 2 = 63504 + 3721 = 67225
Final Answer Finally, we simplify the expression to find the magnitude: ∣ PQ ∣ = 67225 ≈ 259.28 Therefore, the magnitude of the vector PQ is approximately 259.28.
Examples
Vectors are used extensively in video games to represent the position, velocity, and acceleration of objects. For example, if you have two objects in a game, you can use vector subtraction to find the displacement vector between them, and then calculate the magnitude of this vector to find the distance between the objects. This is crucial for collision detection and other game mechanics.
To find the magnitude of the vector \overrightarrow{PQ}, we first calculate the coordinates of points P and Q. We then find the vector \overrightarrow{PQ} by subtracting the coordinates, leading to \overrightarrow{PQ} = (12, 5). The final magnitude is found to be 13.
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