What does this mean? For any vector Vector a find Vector a × Vector a. Explain why (this is cross product stuff). Thanks
Any Vector A cross-producted with itself is zero, because the sine of the angle between the two vectors is zero.
Check my thinking.
Ah, the cross product! It's like the peanut butter to the jelly of vector operations. So, when you have a vector a and you want to find a × a, it's like asking for the square dance of the same partner.
Now, brace yourself for some math fun! When you take the cross product of a vector with itself, you get another vector that's always perpendicular to the original one. It's like turning left and ending up facing forward. So, in this case, Vector a × Vector a will always result in a vector that's orthogonal to Vector a.
But guess what? There's a little more to it. Since Vector a is parallel to itself, the angle between the two is zero. And the sine of zero is zero, my friend! That means the magnitude of Vector a × Vector a is zero. So, the result is a vector that's perpendicular to Vector a, but it doesn't really go anywhere – it's just hanging out at the zero vector party.
Long story short, Vector a × Vector a = the zero vector, and all the vectors are perpendicular and having a blast!
The expression "a × a" represents the cross product of vector a with itself. In order to find a × a, we can follow these steps:
1. Write down the components of vector a. Let's assume vector a has components (a₁, a₂, a₃).
2. Formulate the cross product equation using the determinant notation:
a × a = | i j k |
| a₁ a₂ a₃ |
| a₁ a₂ a₃ |
3. Apply the determinant formula to evaluate the cross product:
a × a = (a₂ * a₃ - a₃ * a₂)i - (a₁ * a₃ - a₃ * a₁)j + (a₁ * a₂ - a₂ * a₁)k
Simplifying further, we get:
a × a = 0i + 0j + 0k
4. The result of a × a is the zero vector [0, 0, 0].
Now, let's explain why the result is zero. In the cross product of a vector with itself, the resulting vector is always a zero vector. This can be understood by examining the properties of the cross product. The cross product of two vectors is perpendicular to both vectors. Since vector a is the same as vector a, the angle between them is zero. In turn, the sine of zero degrees is zero, which means that the magnitude of the cross product is zero. Therefore, the resulting vector is the zero vector [0, 0, 0].
The expression "Vector a × Vector a" represents the cross product of Vector a with itself. In order to find Vector a × Vector a, we first need to understand what a cross product is.
The cross product is an operation that takes two vectors and produces a new vector that is orthogonal (perpendicular) to both of the original vectors. The magnitude of the cross product vector is given by the product of the magnitudes of the original vectors multiplied by the sine of the angle between them. The direction of the cross product vector is determined by the right-hand rule.
However, when calculating the cross product of a vector with itself, the result is always zero. This is because any vector is parallel to itself, and the sine of the angle between two parallel vectors is zero. Therefore, the magnitude of the cross product is zero, and since the direction is orthogonal to both vectors (which are the same in this case), it is also a zero vector.
In summary, for any vector Vector a, the cross product Vector a × Vector a is always the zero vector.
To find this result, follow these steps:
1. Write down Vector a as a set of components. For example, Vector a = (a1, a2, a3).
2. Calculate the cross product of Vector a with itself using the formula for cross product:
Vector a × Vector a = (a2 * a3 - a3 * a2, a3 * a1 - a1 * a3, a1 * a2 - a2 * a1).
3. Simplify the resulting expression for Vector a × Vector a:
(0, 0, 0).
4. The answer is the zero vector, also represented as the null vector or the vector with all components equal to zero: Vector a × Vector a = (0, 0, 0).