Vector A = 2i + 3j + 4k

To find a vector that is perpendicular to vector A, we need to find a vector that has a dot product of zero with vector A. The dot product of two vectors is given by the formula:

A · B = (A_x * B_x) + (A_y * B_y) + (A_z * B_z)

where A_x, A_y, and A_z are the components of vector A, and B_x, B_y, and B_z are the components of vector B.

Let's assume the vector we are looking for is B = xi + yj + zk. We can substitute the values of A and B into the dot product formula:

(A · B) = (2i + 3j + 4k) · (xi + yj + zk)
= (2x) + (3y) + (4z)

Since we want the dot product to be zero, we can set the equation equal to zero:

2x + 3y + 4z = 0

Now we need to find a solution to this equation. We can choose any values for x, y, and z as long as they satisfy the equation. To simplify the process, we can set one of the variables to a known value, and then solve for the other two.

Let's set z = 1. Now the equation becomes:

2x + 3y + 4(1) = 0
2x + 3y + 4 = 0

We can solve this equation for x in terms of y:

2x = -3y - 4
x = (-3y - 4)/2
x = (-3/2)y - 2

So, any vector of the form B = (-3/2)y - 2)i + yj + zk, where y is any real number, will be perpendicular to vector A.

To summarize:
- The vector B = (-3/2)y - 2)i + yj + zk, where y is any real number, is perpendicular to vector A = 2i + 3j + 4k.
- The dot product of vector A and vector B is zero, indicating perpendicularity.
- The equation 2x + 3y + 4z = 0 represents the condition for the dot product to be zero.
- By setting one of the variables (z) to a known value (1), we can solve for the other two variables (x and y).
- The general form of the perpendicular vector B is given by B = (-3/2)y - 2)i + yj + zk.

Dot products of two perpendicular vectors is zero.

Hence

2x + 3y = -4z

In option (d) value of x, y and z is 1, 2 and -2 respectively.

Substitute these values in above equation

2(1) + 3(2) = -4(-2)

2 + 6 = 8

8 = 8