Hey there, math enthusiasts! Today, we're diving into a fun vector problem where we need to find the magnitude and direction angle of a resultant vector. Let's break it down step-by-step so you can conquer similar problems with ease. So, guys, let's get started!
Problem Statement
We are given two vectors: a = (-7, 3) and b = (-2, -12). Our mission, should we choose to accept it (and we do!), is to find the magnitude and direction angle of the vector c, where c = a + b. We have a few options to choose from:
- |c| = 18 ; θ = 135°
- |c| = 18 ; θ = 225°
- |c| = 9√2 ; θ = 135°
- |c| = 9√2 ; θ = 225°
Step 1: Finding Vector c
The first thing we need to do is find the vector c, which is simply the sum of vectors a and b. To add vectors, we add their corresponding components. This means we add the x-components together and the y-components together.
So, c = a + b = (-7, 3) + (-2, -12). Let's do the math:
- x-component of c: -7 + (-2) = -9
- y-component of c: 3 + (-12) = -9
Therefore, c = (-9, -9). Now we have our resultant vector, and we're one step closer to solving the puzzle!
Step 2: Calculating the Magnitude of Vector c
The magnitude of a vector, often denoted by | | (double bars), represents its length. We can calculate the magnitude using the Pythagorean theorem. If a vector has components (x, y), its magnitude |v| is given by:
|v| = √(x² + y²)
In our case, vector c has components (-9, -9). So, let's plug those values into the formula:
|c| = √((-9)² + (-9)²) = √(81 + 81) = √(162)
Now, we can simplify √(162). Notice that 162 is 81 times 2, and 81 is a perfect square (9²). So, we can rewrite it as:
|c| = √(81 * 2) = √81 * √2 = 9√2
Awesome! We found the magnitude of vector c. It's 9√2. Looking at our options, this narrows it down to the last two choices. We're on the right track!
Step 3: Determining the Direction Angle of Vector c
The direction angle (θ) of a vector is the angle it makes with the positive x-axis, measured counterclockwise. We can find this angle using trigonometry, specifically the arctangent function (tan⁻¹), also known as the inverse tangent.
The formula to find the direction angle is:
θ = tan⁻¹(y / x)
Where x and y are the components of the vector. For vector c = (-9, -9), we have x = -9 and y = -9. Let's plug these values into the formula:
θ = tan⁻¹((-9) / (-9)) = tan⁻¹(1)
Now, we know that tan⁻¹(1) gives us a reference angle of 45° (or π/4 radians). However, we need to be careful about the quadrant in which our vector lies. Remember, the arctangent function only gives us angles in the first and fourth quadrants.
Since both the x and y components of vector c are negative, it lies in the third quadrant. To find the correct angle in the third quadrant, we need to add 180° to our reference angle:
θ = 45° + 180° = 225°
Alternatively, you could visualize the vector (-9, -9) on the coordinate plane. It clearly points in the southwest direction, which corresponds to an angle of 225°.
Step 4: The Grand Reveal – The Answer!
We've done all the calculations! We found that the magnitude of vector c is 9√2, and its direction angle is 225°. Now, let's match this with our options. The correct answer is:
- |c| = 9√2 ; θ = 225°
Victory! We solved it! This was a fantastic journey of vector addition, magnitude calculation, and direction angle determination. You guys did awesome!
Key Concepts Revisited
Let's recap the key concepts we used to solve this problem. This will help solidify your understanding and prepare you for future challenges.
- Vector Addition: To add vectors, we simply add their corresponding components. If a = (a₁, a₂) and b = (b₁, b₂), then a + b = (a₁ + b₁, a₂ + b₂).
- Magnitude of a Vector: The magnitude (or length) of a vector v = (x, y) is given by |v| = √(x² + y²). This is based on the Pythagorean theorem.
- Direction Angle of a Vector: The direction angle (θ) is the angle the vector makes with the positive x-axis. We can find it using the arctangent function: θ = tan⁻¹(y / x). Remember to consider the quadrant of the vector to get the correct angle.
Common Pitfalls to Avoid
While solving vector problems, there are a few common mistakes you want to avoid. Being aware of these pitfalls can save you time and frustration.
- Forgetting the Quadrant: As we saw, the arctangent function only gives angles in the first and fourth quadrants. Always visualize the vector to determine the correct quadrant and adjust the angle accordingly.
- Mixing Up Components: Make sure you're adding the correct components when performing vector addition. Double-check your work to avoid simple arithmetic errors.
- Incorrect Magnitude Calculation: Remember to square both components before adding them under the square root. A common mistake is to forget the square.
Practice Makes Perfect
The best way to master vector problems is to practice! Try solving similar problems with different vectors. You can also explore more complex scenarios, such as finding the resultant of three or more vectors.
Real-World Applications
Vectors aren't just abstract mathematical concepts. They have tons of real-world applications in physics, engineering, computer graphics, and more. For example:
- Physics: Vectors are used to represent forces, velocities, and accelerations. Analyzing these vectors helps us understand the motion of objects.
- Engineering: Engineers use vectors to design structures, analyze stress and strain, and control robotic systems.
- Computer Graphics: Vectors are fundamental in computer graphics for representing positions, directions, and transformations of objects in 3D space.
Conclusion
Well, guys, we've reached the end of our vector adventure! We successfully found the magnitude and direction angle of vector c. Remember the key steps: add the vectors, calculate the magnitude using the Pythagorean theorem, and find the direction angle using the arctangent function, paying close attention to the quadrant. Keep practicing, and you'll become a vector master in no time! And remember, math can be fun, especially when you break it down step by step. Keep exploring, keep learning, and I'll catch you in the next mathematical escapade!