If a * b denotes the larger of 'a' and 'b' and if a o b = (a + b) + 3, then write the value of (5) * (10), where * and o are binary operations. Find the magnitude of each of the tw... If a * b denotes the larger of 'a' and 'b' and if a o b = (a + b) + 3, then write the value of (5) * (10), where * and o are binary operations. Find the magnitude of each of the two vectors a and b, having the same magnitude such that the angle between them is 60 degrees, and their scalar product is 9/2. If the matrix [[0, a], [2, 0]] is skew symmetric, find the values of 'a' and 'b'.
Understand the Problem
The question is asking about solving problems related to binary operations, vectors, and matrix symmetry, which are common topics in mathematics. It requires mathematical reasoning and understanding of operations and properties of vectors and matrices.
Answer
The magnitudes are \( |\mathbf{a}| = |\mathbf{b}| = 3 \).
Answer for screen readers
The magnitude of each of the vectors ( \mathbf{a} ) and ( \mathbf{b} ) is 3, or ( |\mathbf{a}| = |\mathbf{b}| = 3 ).
Steps to Solve
- Identify Given Values
We're given two vectors ( \mathbf{a} ) and ( \mathbf{b} ) with the following:
[ |\mathbf{a}| = 3, \quad \theta = 60^\circ, \quad \mathbf{b} \cdot \mathbf{a} = \frac{9}{2} ]
- Use the Cosine Formula for Dot Product
The dot product of two vectors can be expressed as:
[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) ]
Substituting the known values:
[ \frac{9}{2} = 3 |\mathbf{b}| \cos(60^\circ) ]
- Calculate Cosine of the Angle
Since ( \cos(60^\circ) = \frac{1}{2} ), we can replace this in our equation:
[ \frac{9}{2} = 3 |\mathbf{b}| \cdot \frac{1}{2} ]
- Solve for Magnitude of Vector ( \mathbf{b} )
Simplifying the equation gives us:
[ \frac{9}{2} = \frac{3}{2} |\mathbf{b}| ]
Multiplying both sides by 2:
[ 9 = 3 |\mathbf{b}| ]
Now divide by 3:
[ |\mathbf{b}| = 3 ]
The magnitude of each of the vectors ( \mathbf{a} ) and ( \mathbf{b} ) is 3, or ( |\mathbf{a}| = |\mathbf{b}| = 3 ).
More Information
Both vectors have the same magnitude, which is 3 units. This is a specific property in vector operations where two vectors can have the same length even though they may point in different directions.
Tips
- Forgetting the angle in the cosine formula: Always remember to use the angle between the vectors correctly.
- Mixing up magnitudes and directional components: Focus on the magnitudes when applying the cosine formula.
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