If vector A=2i + j & vector B=i - j sketch vectors graphically & find component of vector A along vector B & perpendicular to vector B.

To tackle the problem of sketching vectors A and B, as well as finding the components of vector A along and perpendicular to vector B, let's break it down step by step.

Sketching the Vectors

First, we need to visualize the vectors. Vector A is given as 2i + j, which means it has a horizontal component of 2 and a vertical component of 1. Vector B is i - j, indicating a horizontal component of 1 and a vertical component of -1.

Plotting Vector A

• Start at the origin (0,0).
• Move 2 units to the right (along the x-axis) for the i component.
• Then, move 1 unit up (along the y-axis) for the j component.

So, the endpoint of vector A will be at the coordinates (2, 1).

Plotting Vector B

• Again, start at the origin (0,0).
• Move 1 unit to the right for the i component.
• Then, move 1 unit down for the -j component.

This places the endpoint of vector B at the coordinates (1, -1).

Finding Components of Vector A

Next, we need to find the components of vector A along vector B and perpendicular to vector B. To do this, we first need to determine the unit vector of B.

Calculating the Unit Vector of B

The magnitude of vector B can be calculated using the formula:

|B| = √(B_x² + B_y²)

Substituting the components of B:

|B| = √(1² + (-1)²) = √(1 + 1) = √2

Now, the unit vector of B, denoted as ûB, is:

ûB = B / |B| = (1/√2)i + (-1/√2)j

Component of A Along B

The component of vector A along vector B can be found using the dot product:

A along B = (A · ûB) * ûB

First, we calculate the dot product A · B:

A · B = (2i + j) · (i - j) = 2(1) + 1(-1) = 2 - 1 = 1

Now, we find the component of A along B:

A along B = (1) * (1/√2 i - 1/√2 j) = (1/√2)i - (1/√2)j

Component of A Perpendicular to B

To find the component of A that is perpendicular to B, we can use the formula:

A perpendicular to B = A - (A along B)

Substituting the values we have:

A perpendicular to B = (2i + j) - ((1/√2)i - (1/√2)j)

Calculating this gives:

A perpendicular to B = (2 - 1/√2)i + (1 + 1/√2)j

Summary of Results

In summary, we have:

• Vector A: 2i + j
• Vector B: i - j
• Component of A along B: (1/√2)i - (1/√2)j
• Component of A perpendicular to B: (2 - 1/√2)i + (1 + 1/√2)j

By visualizing the vectors and performing these calculations, we gain a clearer understanding of their relationships in space. If you have any further questions or need clarification on any of these steps, feel free to ask!