Vector Projection Find Proj_{u}(v + W) Given U, V, And W

Jul 23, 2025 by ADMIN 57 views

Finding the Projection of a Vector Sum onto Another Vector

Hey there, math enthusiasts! Today, we're diving into a cool vector problem that involves finding the projection of the sum of two vectors onto another vector. It sounds a bit complex, but we'll break it down step by step so it's super easy to follow. So, let's get started and learn how to tackle this projection problem like pros!

Problem Statement

Okay, so here's the problem we're going to solve. We're given three vectors:

u = -i + j
v = 6i - 2j
w = -8j

Our mission, should we choose to accept it (and we do!), is to find the projection of the vector sum (v + w) onto the vector u. In mathematical notation, we need to find proj_u(v + w).

Understanding Vector Projection

Before we jump into the calculations, let's quickly recap what vector projection actually means. Imagine you have two vectors, say vector a and vector b. The projection of vector b onto vector a, written as proj_a(b), is essentially the shadow of vector b that falls onto vector a. It's the component of b that lies in the direction of a. Think of it like shining a light perpendicular to vector a; the shadow cast by vector b is the projection.

The Formula for Vector Projection

To calculate the projection mathematically, we use the following formula:

proj_a(b) = ((a · b) / ||a||²) * a

Where:

a · b is the dot product of vectors a and b.
||a|| is the magnitude (or length) of vector a.
||a||² is the square of the magnitude of vector a.

Now that we've refreshed our understanding of vector projection, let's roll up our sleeves and solve the problem!

Step-by-Step Solution

Step 1: Find the Vector Sum (v + w)

First things first, we need to find the sum of vectors v and w. This is a straightforward process of adding the corresponding components:

v + w = (6i - 2j) + (-8j) = 6i + (-2 - 8)j = 6i - 10j

So, the vector sum v + w is 6i - 10j. Easy peasy, right?

Step 2: Calculate the Dot Product u · (v + w)

Next up, we need to calculate the dot product of vector u and the vector sum (v + w). Remember, the dot product of two vectors a = a₁i + a₂j and b = b₁i + b₂j is given by:

a · b = a₁b₁ + a₂b₂

Applying this to our vectors u = -i + j and (v + w) = 6i - 10j, we get:

u · (v + w) = (-1)(6) + (1)(-10) = -6 - 10 = -16

So, the dot product u · (v + w) is -16.

Step 3: Calculate the Magnitude Squared of u (||u||²)

Now, we need to find the magnitude squared of vector u. The magnitude of a vector a = a₁i + a₂j is given by:

||a|| = √(a₁² + a₂²)

And the magnitude squared is simply the square of the magnitude:

||a||² = a₁² + a₂²

For our vector u = -i + j, the magnitude squared is:

||u||² = (-1)² + (1)² = 1 + 1 = 2

Therefore, the magnitude squared of u is 2.

Step 4: Calculate the Projection proj_u(v + w)

Alright, we've got all the pieces of the puzzle! Now we can finally calculate the projection of (v + w) onto u using the formula:

proj_u(v + w) = ((u · (v + w)) / ||u||²) * u

Plugging in the values we calculated, we get:

proj_u(v + w) = (-16 / 2) * (-i + j) = -8 * (-i + j) = 8i - 8j

And there you have it! The projection of (v + w) onto u is 8i - 8j.

Final Answer

The projection of the vector sum (v + w) onto the vector u is:

proj_u(v + w) = 8i - 8j

So, if you were asked to type your answer in terms of i and j, you'd confidently write 8i - 8j.

Key Concepts and Takeaways

Let's quickly recap the key concepts we've covered in this problem:

Vector Sum: Adding vectors involves adding their corresponding components.
Dot Product: The dot product of two vectors is a scalar value calculated by multiplying corresponding components and summing the results.
Magnitude of a Vector: The magnitude (or length) of a vector is calculated using the Pythagorean theorem.
Vector Projection: The projection of one vector onto another represents the component of the first vector that lies in the direction of the second vector.
Projection Formula: The formula proj_a(b) = ((a · b) / ||a||²) * a is the key to calculating vector projections.

By understanding these concepts and following the step-by-step solution, you can confidently tackle similar vector projection problems. Remember, practice makes perfect, so try solving more problems to solidify your understanding.

Common Mistakes to Avoid

When working with vector projections, there are a few common mistakes that students often make. Let's highlight these so you can steer clear of them.

Forgetting the Order of Operations: Make sure you calculate the dot product and magnitude squared before multiplying by the vector. The order of operations matters!
Incorrectly Calculating the Dot Product: Double-check your calculations when finding the dot product. A simple arithmetic error can throw off your entire solution.
Mixing Up Vectors: Be careful to use the correct vectors in the projection formula. It's easy to get mixed up, especially when dealing with multiple vectors.
Forgetting to Square the Magnitude: Remember that the denominator in the projection formula is the magnitude squared, not just the magnitude.

By being mindful of these common pitfalls, you can increase your accuracy and avoid unnecessary errors. Always double-check your work and take your time.

Practice Problems

To really master vector projections, it's essential to practice! Here are a few practice problems you can try:

Given u = 2i + 3j, v = i - j, and w = 4j, find proj_u(v + w).
Given a = -3i + 2j, b = 5i + j, and c = -2i - 3j, find proj_a(b + c).
Find the projection of the vector 3i - 4j onto the vector i + j.

Try solving these problems on your own, and then check your answers with online resources or your instructor. The more you practice, the more confident you'll become!

Conclusion

So, there you have it, guys! We've successfully found the projection of a vector sum onto another vector. We've broken down the problem step by step, discussed the key concepts, and even highlighted common mistakes to avoid. With a solid understanding of vector projections and some practice, you'll be able to solve these types of problems with ease.

Keep practicing, keep learning, and keep exploring the fascinating world of vectors! Until next time, happy math-ing!