Find as unit vector in the direction of the resultant of vectors A=2i-3j+k, B=i+j+2k and C=3i-2j+4k

2i-3j+k + i+j+2k + 3i-2j+4k

= 6i -4j + 7k

magnitude = sqrt(36+16+49) = sqrt(101)

your unit vector is < 6/sqrt(101) , -4/sqrt(101), 7/sqrt(101) >

6i-2j+7k

To find the unit vector in the direction of the resultant, we need to compute the resultant vector and then normalize it.

1. Compute the resultant vector R:
R = A + B + C

Given:
A = 2i - 3j + k
B = i + j + 2k
C = 3i - 2j + 4k

Substitute the values:
R = (2i - 3j + k) + (i + j + 2k) + (3i - 2j + 4k)
= 2i - 3j + k + i + j + 2k + 3i - 2j + 4k
= (2 + 1 + 3)i + (-3 + 1 - 2)j + (1 + 2 + 4)k
= 6i - 4j + 7k

2. Compute the magnitude of R:
||R|| = √(R_x^2 + R_y^2 + R_z^2)

Given:
R = 6i - 4j + 7k

Substitute the values:
||R|| = √((6)^2 + (-4)^2 + (7)^2)
= √(36 + 16 + 49)
= √101

3. Normalize the vector R to find the unit vector in the direction of the resultant:
To find the unit vector, we divide the vector R by its magnitude.

Unit vector in the direction of R = R / ||R||

Given:
R = 6i - 4j + 7k
||R|| = √101

Substitute the values:
Unit vector in the direction of R = (6i - 4j + 7k) / √101

To find the unit vector in the direction of the resultant of vectors A, B, and C, we need to first find the sum of these vectors and then normalize it.

1. Start by adding the vectors A, B, and C together to find their resultant vector R:
R = A + B + C

R = (2i - 3j + k) + (i + j + 2k) + (3i - 2j + 4k)

Simplifying the sum:
R = 6i - 4j + 7k

2. To normalize the resultant vector R, divide it by its magnitude. The magnitude of a vector is calculated using the formula:
||R|| = sqrt(Rx^2 + Ry^2 + Rz^2), where Rx, Ry, and Rz are the components of R.

Magnitude of R:
||R|| = sqrt((6)^2 + (-4)^2 + (7)^2)

||R|| = sqrt(36 + 16 + 49)

||R|| = sqrt(101)

3. Finally, divide the resultant vector R by its magnitude to obtain the unit vector in the direction of R:
Unit vector U = R / ||R||

U = (6i - 4j + 7k) / sqrt(101)

Now you have the unit vector U in the direction of the resultant of vectors A, B, and C.