Two vectors of magnitudes 10 units and 15 units are acting at a point. The magnitude of their resultant is 20 units. Find the angle between them.
The parallelogram of forces yields a triangle with vectors 10,15 and 20.
where 20 is the resultant.
The angle between the two vector is the supplement of the angle (between vectors 10 and 15) solved by the cosine rule.
Draw the parallelogram of forces to understand why it is the supplement.
So
cos((φ)=sqrt(10²+15²-20²)/(2*10*15))
and the angle between vectors 10 and 15 is 180°-φ.
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To find the angle between two vectors, we can use the dot product formula and inverse cosine function. Here's how to calculate it step by step:
1. Start by considering the vectors A and B, with magnitudes 10 units and 15 units respectively.
2. Let the angle between these vectors be θ.
3. The dot product of two vectors A and B is defined as A · B = |A| |B| cos(θ), where |A| and |B| are the magnitudes of vectors A and B.
4. Knowing that the resultant magnitude is 20 units, we can write |A| |B| cos(θ) = 20.
5. Substitute the given magnitudes: (10)(15) cos(θ) = 20.
6. Simplify: 150 cos(θ) = 20.
7. Divide both sides by 150: cos(θ) = 20/150.
8. Calculate the value of cos(θ) using a calculator: cos(θ) ≈ 0.1333.
9. Now, take the inverse cosine (cos^(-1)) of 0.1333 to find the angle θ.
θ ≈ cos^(-1)(0.1333).
10. Evaluate the inverse cosine using a calculator: θ ≈ 82.69 degrees (rounded to two decimal places).
Therefore, the angle between the two vectors is approximately 82.69 degrees.