Answer:
a) The scalar product of vectors a and b is 26.01 units.
b) The magnitude of the vector product axb is 64.59 units.
Step-by-step explanation:
To find the scalar product (also known as the dot product) of two vectors a and b, you can use the following formula:
(a · b) = |a| * |b| * cos(θ)
where:
(a · b) represents the dot product of vectors a and b.
|a| is the magnitude (length) of vector a.
|b| is the magnitude (length) of vector b.
θ is the angle between vectors a and b.
In this case, you're given:
|a| = 15 units
|b| = 4.6 units
θ = 75°
Let's calculate the scalar product (a · b):
(a · b) = 15 * 4.6 * cos(75°)
Now, we need to find the value of cos(75°). We can use the trigonometric identity:
cos(75°) = cos(45° + 30°)
Using the sum of angles formula for cosine:
cos(45° + 30°) = cos(45°)cos(30°) - sin(45°)sin(30°)
We know that cos(45°) and sin(45°) are both equal to 1/√2, and cos(30°) and sin(30°) are √3/2 and 1/2, respectively.
cos(75°) = (1/√2) * (√3/2) - (1/√2) * (1/2)
cos(75°) = (√3/2√2) - (1/2√2)
cos(75°) = (√3 - 1) / (2√2)
Now, plug this value back into the dot product formula:
(a · b) = 15 * 4.6 * [(√3 - 1) / (2√2)]
(a · b) ≈ 15 * 4.6 * 0.366 = 26.01
So, the scalar product of vectors a and b is 26.01 units.
Now, let's find the magnitude of the vector product (axb). The magnitude of the vector product is given by:
|axb| = |a| * |b| * sin(θ)
We already have the values for |a| and |b| from the previous calculation. Now, we'll use sin(θ), where θ is 75°:
|axb| = 15 * 4.6 * sin(75°)
To find sin(75°), we can again use trigonometric identities:
sin(75°) = sin(45° + 30°)
Using the sum of angles formula for sine:
sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)
sin(45°) and cos(45°) are both equal to 1/√2, and sin(30°) and cos(30°) are 1/2 and √3/2, respectively.
sin(75°) = (1/√2) * (1/2) + (1/√2) * (√3/2)
sin(75°) = (1/2√2) + (√3/2√2)
sin(75°) = (√3 + 1) / (2√2)
Now, calculate the magnitude of the vector product:
|axb| = 15 * 4.6 * [(√3 + 1) / (2√2)]
|axb| ≈ 15 * 4.6 * 0.933 = 64.59
So, the magnitude of the vector product axb is 64.59 units.