Vector Algebra

To find , express the given vector in terms of , , and their cross product, then solve for the coefficients.

Given:

Step 1: Calculate and

Step 2: Calculate

Step 3: Write the equation for components:
Set equal to :

Step 4: Solve equations (1), (2), and (3)
From (1): Substitute into (2): Substitute and into (3): Combine constants: Combine terms (common denominator 15): Equation becomes: Multiply both sides by 3:

Final Answer:

Vector Expressions

Given vectors:

Cross Product

Computing the determinant:

Expanding along the first row:

Dot Product with

Solving for .

Substituting into the Plane Equation

Solving for :

Final Answer: 5