Vectors

To find the volume of the parallelepiped formed by vectors a, b, and c, we need to calculate the scalar triple product, defined as:

Volume = a ⋅ (b × c).

Given the vectors:

• a = î + 2ĵ - k̂,
• b = 2î - ĵ + k̂,
• c = î + ĵ + k̂.

First, compute the cross product b × c.

The formula for the cross product of two vectors u and v is:

u × v = (u₂v₃ - u₃v₂)î + (u₃v₁ - u₁v₃)ĵ + (u₁v₂ - u₂v₁)k̂.

Applying this formula:

b × c = (−1×1 − 1×1)î + (1×1 − 2×1)ĵ + (2×1 − (−1)×1)k̂.

This simplifies to:

b × c = (−1 − 1)î + (1 − 2)ĵ + (2 + 1)k̂ = −2î − 1ĵ + 3k̂.

Now, compute the dot product a ⋅ (b × c).

The dot product d ⋅ v is given by:

d ⋅ v = d₁v₁ + d₂v₂ + d₃v₃.

Using the values:

a = î + 2ĵ - k̂, b × c = −2î − 1ĵ + 3k̂.

a ⋅ (b × c) = 1×(−2) + 2×(−1) + (−1)×3,

This calculates to:

−2 − 2 − 3 = −7.

Hence, the volume of the parallelepiped is -7.