What is the component of 3i+4j along i+j ?

In both physics and mathematics, vectors are essential tools for representing quantities that have magnitude and direction. One of the interesting operations involving vectors is finding the component of one vector along another. Not only is this concept important in theory, but it’s also super useful in practical areas like engineering, computer science, and physics.

In this post, we'll learn to find the component of the vector 3i + 4j along the vector i + j . With this example you will get to know solving such type of questions.

What is the component of 3i+4j along i+j ?

Understanding Vectors and Components

First of all, let's break down the vectors in the given question:

3i + 4j is a vector in Cartesian coordinate system, where i and j are unit vectors along x-axis and y-axis, respectively. The vector 3i + 4j has a horizontal component of 3 units along x-axis and a vertical component of 4 units along y-axis.

i + j is another vector with equal components, along x and y axes. This represents a 45-degree angle with the positive x-axis.

To find the components of 3i + 4j along i + j, we're essentially projecting one vector on another.

The Mathematical Formula

The component of vector A along vector B is given by the dot product of A and unit vector in the direction of B. The formula is:

Component of A along B= (A.B) /∣B∣

Where:

A · B is the dot product of vectors A and B

|B|is the magnitude (length) of vector B

Step-by-Step Solution

Let's apply this to vectors A = 3i + 4j and B = i + j

1. Dot Product of A and B:

The dot product A · B is calculated as follows:

          A⋅B=(3i+4j)⋅(i+j)

Expanding the dot product:

          A⋅B=(3×1)+(4×1)=3+4=7

2. Magnitude of B:

The magnitude |B| of vector B = i + j is:

          ∣B∣= sqrt[(1)2+(1)2] = sqrt(2)

3. Component of A along B:

Finally, the component of 3i + 4j along i + j is:

          Component=7/sqrt(2)=(7×sqrt(2))/2

So, the component of 3i + 4j along i + j is (7×sqrt(2)) / 2.

Conclusion

The component of a vector with another vector is an important concept in vector analysis. It is applicable in physics (such as resolving forces) and in other scientific fields. In this example, we found that the component of 3i + 4j along i + j is (7×sqrt(2)) / 2. This value gives us the magnitude of the projection of one vector on another, which shows that how much of vector 3i + 4j lies in the direction of i + j .

By understanding and calculating vector components you can greatly enhance your ability to solve complex problems. 

Practice Questions 

Here are some questions that you should solve for practice and to understood better.

1. How would you find the component of the vector 5i + 2j along the vector i - j?"
2. "If the vector A is 4i - 3j and vector B is 2i + 2j, what is the component of A along B?"
3. "What does it mean if the component of one vector along another is zero?"
4. "Can you calculate the component of 7i + 6j along the vector 3i + 4j?"
5. "How would the component of 2i + 3j along i + j change if the vectors were multiplied by a scalar?"

If you have any other doubts you can freely ask me in comment sections 🙂

Impact-Site-Verification: b5d17d4e-d512-4b2f-a580-4b284b98c5d8