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Question 1179249: Find value of the folllwing:
(a) (a. b) + (a x b)²
(b) (a. b) + (a x 2b)²
.
Note: Here . means dot product and x means cross or vector product.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve these vector operations:
**(a) (a · b) + |a × b|²**
* **Dot Product (a · b):**
* a · b = |a||b|cos(θ), where θ is the angle between vectors a and b.
* **Cross Product (a × b):**
* |a × b| = |a||b|sin(θ)
* |a × b|² = |a|²|b|²sin²(θ)
Now, let's substitute these into the given expression:
(a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)
However, we can simplify this further using a trigonometric identity: sin²(θ) + cos²(θ) = 1.
* |a · b|² = |a|²|b|²cos²(θ)
Therefore,
(a · b)² + |a × b|² = |a|²|b|²cos²(θ) + |a|²|b|²sin²(θ)
= |a|²|b|²(cos²(θ) + sin²(θ))
= |a|²|b|²(1)
= |a|²|b|²
So, the original expression is not (a.b)^2 + |axb|^2 , but (a.b) + |axb|^2.
Thus,
(a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)
This cannot be further simplified without knowing the vectors a and b.
**(b) (a · b) + |a × 2b|²**
* **Dot Product (a · b):**
* a · b = |a||b|cos(θ)
* **Cross Product (a × 2b):**
* a × 2b = 2(a × b)
* |a × 2b| = 2|a × b| = 2|a||b|sin(θ)
* |a × 2b|² = 4|a|²|b|²sin²(θ)
Now, substitute these into the given expression:
(a · b) + |a × 2b|² = |a||b|cos(θ) + 4|a|²|b|²sin²(θ)
Again, this expression cannot be further simplified without knowing the vectors a and b.
**Summary:**
* (a · b) + |a × b|² = |a||b|cos(θ) + |a|²|b|²sin²(θ)
* (a · b) + |a × 2b|² = |a||b|cos(θ) + 4|a|²|b|²sin²(θ)
If the question had been (a.b)^2 + |axb|^2 , then the answer would be |a|^2|b|^2.
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