Given:
Curve: x = 1 - t, y = 1, z = 2t - 5
Vector: V = 2i + j + 2k

To find:
Components of velocity and acceleration at t = 1 in the direction of vector V.

Solution:

Step 1: Find the velocity vector
The velocity vector can be found by differentiating the position vector with respect to time.

Given position vector: r = xi + yj + zk
Differentiating with respect to time:
v = dr/dt = dx/dt i + dy/dt j + dz/dt k

Given x = 1 - t, y = 1, z = 2t - 5
Differentiating x with respect to t:
dx/dt = -1
Differentiating y with respect to t:
dy/dt = 0
Differentiating z with respect to t:
dz/dt = 2

Substituting the values into the velocity vector equation:
v = -1i + 0j + 2k
v = -i + 2k

Therefore, the velocity vector is v = -i + 2k.

Step 2: Find the acceleration vector
The acceleration vector can be found by differentiating the velocity vector with respect to time.

Given velocity vector: v = -i + 2k
Differentiating with respect to time:
a = dv/dt = d^2x/dt^2 i + d^2y/dt^2 j + d^2z/dt^2 k

Differentiating x with respect to t:
d^2x/dt^2 = 0
Differentiating y with respect to t:
d^2y/dt^2 = 0
Differentiating z with respect to t:
d^2z/dt^2 = 0

Substituting the values into the acceleration vector equation:
a = 0i + 0j + 0k
a = 0

Therefore, the acceleration vector is a = 0.

Step 3: Find the components of velocity and acceleration at t = 1 in the direction of vector V
To find the components of velocity and acceleration in the direction of vector V, we need to find the dot product of V with the velocity and acceleration vectors.

Given V = 2i + j + 2k and velocity vector v = -i + 2k, the dot product is given by:
V . v = (2i + j + 2k) . (-i + 2k)

Expanding the dot product:
V . v = 2(-1) + 1(0) + 2(2)
V . v = -2 + 0 + 4
V . v = 2

Similarly, given V = 2i + j + 2k and acceleration vector a = 0, the dot product is given by:
V . a = (2i + j + 2k) . (0i + 0j + 0k)

Expanding the dot product:
V . a = 2(0) + 1(0) + 2(0)
V . a = 0

Therefore, the component of velocity