Find the scalar product of the vectors ai+bj and bi-aj . Where a and b are arbitrary constants. What's the angle between the two vectors?
Well, well, prepare yourself for some vector comedy! Let's start by finding the scalar product (also known as the dot product) of the vectors ai+bj and bi-aj.
The scalar product between two vectors is calculated by multiplying the corresponding components and adding them up. So, let's do some math magic!
(ai + bj) • (bi - aj) = (ab)i² + (bj)² - (ab)j² - (aj)²
Since i² = j² = -1, we can simplify it further:
(ab)i² + (bj)² - (ab)j² - (aj)² = -ab - ab - b² + a² = -2ab - b² + a²
There you have it, the scalar product of the vectors ai+bj and bi-aj is -2ab - b² + a². That's a mouthful, isn't it?
Now, moving on to the angle between these two vectors. To find the angle, we need to use the formula:
cosθ = (ai + bj) • (bi - aj) / ||ai + bj|| * ||bi - aj||
But hey, since we already have the scalar product, we can make this easier. The magnitude of a vector is simply the square root of its scalar product with itself. So, we can say:
cosθ = (-2ab - b² + a²) / √((a² + b²)(a² + b²))
Now you might be thinking, "What now, Clown Bot? I'm not a fan of trigonometry!" Well, fear not! If you want to find the angle in degrees, you can use the inverse cosine (arccos) function.
θ = arccos(cosθ)
And there you go! You now have the angle between those two vectors, expressed humorously, of course.
To find the scalar product of the vectors ai + bj and bi - aj, we'll use the formula for the dot product (or scalar product) of two vectors:
𝐀 ⋅ 𝐁 = |𝐀| |𝐁| cos(θ)
Here, 𝐀 and 𝐁 represent the two vectors, |𝐀| and |𝐁| are their magnitudes, and θ is the angle between the two vectors.
Let's calculate the dot product of the two vectors:
(ai + bj) ⋅ (bi - aj)
= (a * b) + (b * (-a))
= a * b - a * b
= 0
The dot product of the vectors is zero. This suggests that the vectors are perpendicular to each other, as the dot product of two vectors is zero when they are perpendicular.
To find the angle between the vectors, we can use the following formula, based on the dot product:
cos(θ) = 𝐀 ⋅ 𝐁 / (|𝐀| |𝐁|)
Plugging in the values we've found:
cos(θ) = 0 / (|𝐀| |𝐁|)
cos(θ) = 0
Since the cosine of the angle between the vectors is zero, the angle θ must be 90 degrees or π/2 radians. This confirms that the two vectors are indeed perpendicular to each other.