i travel 1 km east, than 1 km north, then climb 400m straight up a cliff. what is my distance from my starting point?

a= 3i+2j+4k
b= i-3j-4k

To calculate your distance from the starting point, we can use the formula for the magnitude of a vector. The formula is:

|c| = √(cx^2 + cy^2 + cz^2)

Where cx, cy, and cz are the components of vector c.

Let's calculate the two vectors step by step:

Vector a = 3i + 2j + 4k
cx = 3, cy = 2, cz = 4
|a| = √(3^2 + 2^2 + 4^2) = √(9 + 4 + 16) = √29

Vector b = i - 3j - 4k
cx = 1, cy = -3, cz = -4
|b| = √(1^2 + (-3)^2 + (-4)^2) = √(1 + 9 + 16) = √26

Now, let's add these two vectors:

c = a + b = (3i + 2j + 4k) + (i - 3j - 4k)
c = 4i - j

cx = 4, cy = -1, cz = 0
|c| = √(4^2 + (-1)^2 + 0^2) = √(16 + 1 + 0) = √17

Therefore, your distance from the starting point is √17 units.

To find the distance from your starting point, we can use the concept of vector addition.

First, let's consider the displacement vectors for each part of your journey:
- The vector for traveling 1 km east is given by a = 1i.
- The vector for traveling 1 km north is given by b = 1j.
- The vector for climbing 400m straight up the cliff is given by c = 0i + 0j + 0.4k.

Now, we can add these vectors to find the total displacement vector:
- The displacement vector, d, is given by d = a + b + c.
d = (1 + 0 + 0)i + (0 + 1 + 0)j + (0 + 0 + 0.4)k
= 1i + 1j + 0.4k

To find the distance from the starting point, we need to find the magnitude of the displacement vector, which can be calculated using the Pythagorean theorem in three-dimensional space.

The magnitude of a vector d is given by |d| = sqrt(dx^2 + dy^2 + dz^2), where dx, dy, and dz are the components of the vector.

In our case, |d| = sqrt(1^2 + 1^2 + 0.4^2)
= sqrt(1 + 1 + 0.16)
= sqrt(2.16)
≈ 1.47 km

Therefore, your distance from the starting point is approximately 1.47 km.