# Write <-5,-7,-3> = vector v_1 + vector v_2, where vector v_1 is parallel to <8,5,-5> while vector v_2 is perpendicular to <8,5,-5>.What are vector v_1 and vector v_2 equal to?

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### 1 Answer

You need to write the vector `bar v_1` such that:

`bar v_1 = x_1bar i + y_1 bar j + z_1 bar k`

Since the problem provides the information that the vector `bar v_1` is parallel to the vector `<8,5,-5>` yields:

`bar v_1 = 8a bar i + 5a bar j - 5a bar k`

You need to write the vector `bar v_2` such that:

`bar v_1 = x_2bar i + y_2 bar j + z_2 bar k`

Since the problem provides the information that the vector `bar v_2 ` is perpendicular to `<8,5,-5>` yields:

`8x_2 + 5y_2 -5z_2 = 0`

The problem provides the following equation, such that:

`8a bar i + 5a bar j - 5a bar k + x_2bar i + y_2 bar j + z_2 bar k = -5 bar i - 7 bar j - 3 bar k`

Equating the coefficients of like terms yields:

`8a + x_2 = -5 => x_2 = -5 - 8a`

`5a + y_2 = -7 => y_2 = -7 - 5a`

`-5a + z_2 = -3 =>z_2 = -3 + 5a`

Substituting `x_2 = -5 - 8a, y_2 = -7 - 5a, z_2 = -3 + 5` a in equation `8x_2 + 5y_2 -5z_2 = 0` yields:

`8(-5 - 8a) + 5(-7 - 5a) -5(-3 + 5a) = 0`

`-40 - 64a - 35 - 25a + 15 - 25a = 0`

`-114a = 60 => a = -60/114 => a = -30/57 => a = -10/19`

You may find now `bar v_1` such that:

`bar v_1 = (-80/19) bar i - (50/19) bar j + (50/19) bar k`

You may find now `bar v_2` such that:

`bar v_2 = (-5 + 80/19) bar i + (-7 + 50/19) bar j + (-3 - 50/19) bar k`

`bar v_2 = (-15/19)bar i - (83/19) bar j - (107/19) bar k`

**Hence, evaluating the vectors `bar v_1` and `bar v_2` yields` bar v_1 = (-80/19) bar i - (50/19) bar j + (50/19) bar k` and `bar v_2 = (-15/19)bar i - (83/19) bar j - (107/19) bar k` .**