# Given M1(1,-2,3), M2(-3,5,-2) write the vector form and parametric form of the equation of the line M1M2.

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### 1 Answer

We'll write the vector form of the equation of the line M1M2.

r = r1 + t(r2 - r1) (*),

where r1 and r2 are the vectors of position of the points M1 and M2.

r1 = xM1*i + yM1*j + zM1*k

We'll substitute the coordinates of M1:

r1 = 1*i + (-2)*j + 3*k

r1 = i - 2j + 3k (1)

We'll write the equation of the vector r2:

r2 = xM2*i + yM2*j + zM2*k

r2 = -3*i + 5*j + (-2)*k

r2 = -3i + 5j - 2k (2)

We'll compute the difference:

r2 - r1 = (-3-1)i + (5+2)j + (-2-3)k

r2 - r1 = -4i + 7j - 5k (3)

We'll substitute (1), (2), (3) in (*):

r = i - 2j + 3k + t( -4i + 7j - 5k)

The vector form of the equation of the line that passes through M1 and M2 is:

**r = i - 2j + 3k + t( -4i + 7j - 5k)**

Knowing the vector form of the equation, we'll write the **parametric form**:

**x = 1 - 4t**

**y = -2 + 7t**

**z = 3 - 5t**