For any vector Vector a find Vector a × Vector a. Explain your result.
Well, Vector a × Vector a is not really a thing. It's like asking me to find the square root of a clown's nose. It's just not possible! When you try to calculate the cross product of a vector with itself, you end up with a big, fat zero. So, the answer to Vector a × Vector a is a resounding "nada"!
To find the cross product of vector a with itself (Vector a × Vector a), we follow these steps:
Step 1: Write the components of vector a.
Let's assume vector a has the components: a = (a₁, a₂, a₃).
Step 2: Write the formula for the cross product.
The cross product of two vectors, such as Vector a × Vector a, is given by the formula:
a × a = (a₂ * a₃ - a₃ * a₂) i + (a₃ * a₁ - a₁ * a₃) j + (a₁ * a₂ - a₂ * a₁) k
Step 3: Calculate the cross product.
Substituting the components of vector a into the formula, we have:
a × a = ((a₂ * a₃ - a₃ * a₂) i) + ((a₃ * a₁ - a₁ * a₃) j) + ((a₁ * a₂ - a₂ * a₁) k)
Step 4: Simplify the calculations.
Since we are finding the cross product of vector a with itself, each component will be multiplied by itself. Therefore, the cross product simplifies to:
a × a = (0 i) + (0 j) + (0 k)
Step 5: Finalize the solution.
The resulting cross product is:
a × a = 0
Explanation:
The cross product of vector a with itself is always zero. This is because the cross product represents a vector perpendicular to both vectors being multiplied. Since Vector a is being multiplied by itself, the resulting vector is parallel to Vector a, making it zero.
To find the cross product of a vector with itself, we need to use the formula for cross product:
Vector a × Vector b = ||a|| ||b|| sin(θ) n
where ||a|| and ||b|| are the magnitudes of the two vectors, θ is the angle between them, and n is the unit normal to the plane containing a and b.
In this case, we need to find Vector a × Vector a, which means Vector a is both vectors in the formula. This implies that ||a|| = ||a||, which is simply the magnitude of Vector a.
Since the angle between Vector a and itself is 0 degrees, sin(θ) becomes 0. Therefore, the cross product of Vector a with itself is zero:
Vector a × Vector a = 0
In conclusion, the cross product of a vector with itself is always zero because the sine of the angle between the two identical vectors is zero.