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The angle between two vectors a = i + 2j – k and b = 2i + j + k is
- a)30 deg
- b)45 deg
- c)60 deg
- d)90 deg
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The angle between two vectors a = i + 2j – k and b = 2i + j + k ...
Angle between two vectors:
θ = 60°
θ = 60°
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The angle between two vectors a = i + 2j – k and b = 2i + j + k ...
Angle between two vectors a and b
Vectors a = i + 2j - k and b = 2i + j + k can be represented as:
a = <1, 2,="" -1="">
b = <2, 1,="" 1="">
Dot product
To find the angle between the two vectors, we first need to calculate the dot product of vectors a and b. The dot product of two vectors a and b is given by the formula:
a · b = |a| |b| cos(theta)
Where |a| and |b| are the magnitudes of vectors a and b respectively, and theta is the angle between the two vectors.
Magnitude of vectors
The magnitude of a vector is given by the formula:
|a| = sqrt(a1^2 + a2^2 + a3^2)
Calculating the magnitudes of vectors a and b:
|a| = sqrt(1^2 + 2^2 + (-1)^2) = sqrt(6)
|b| = sqrt(2^2 + 1^2 + 1^2) = sqrt(6)
Calculating the dot product
Now, we can calculate the dot product of vectors a and b:
a · b = <1, 2,="" -1=""> · <2, 1,="" 1=""> = 1*2 + 2*1 + (-1)*1 = 1 + 2 - 1 = 2
Calculating the angle
Substitute the values of dot product, |a| and |b| in the dot product formula:
2 = sqrt(6) * sqrt(6) * cos(theta)
2 = 6 * cos(theta)
cos(theta) = 2/6 = 1/3
Therefore, theta = cos^(-1)(1/3) ≈ 70.53 degrees
The angle between vectors a and b is approximately 70.53 degrees. Since this value is not one of the options provided, the closest option is 60 degrees.
Vectors a = i + 2j - k and b = 2i + j + k can be represented as:
a = <1, 2,="" -1="">
b = <2, 1,="" 1="">
Dot product
To find the angle between the two vectors, we first need to calculate the dot product of vectors a and b. The dot product of two vectors a and b is given by the formula:
a · b = |a| |b| cos(theta)
Where |a| and |b| are the magnitudes of vectors a and b respectively, and theta is the angle between the two vectors.
Magnitude of vectors
The magnitude of a vector is given by the formula:
|a| = sqrt(a1^2 + a2^2 + a3^2)
Calculating the magnitudes of vectors a and b:
|a| = sqrt(1^2 + 2^2 + (-1)^2) = sqrt(6)
|b| = sqrt(2^2 + 1^2 + 1^2) = sqrt(6)
Calculating the dot product
Now, we can calculate the dot product of vectors a and b:
a · b = <1, 2,="" -1=""> · <2, 1,="" 1=""> = 1*2 + 2*1 + (-1)*1 = 1 + 2 - 1 = 2
Calculating the angle
Substitute the values of dot product, |a| and |b| in the dot product formula:
2 = sqrt(6) * sqrt(6) * cos(theta)
2 = 6 * cos(theta)
cos(theta) = 2/6 = 1/3
Therefore, theta = cos^(-1)(1/3) ≈ 70.53 degrees
The angle between vectors a and b is approximately 70.53 degrees. Since this value is not one of the options provided, the closest option is 60 degrees.
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The angle between two vectors a = i + 2j – k and b = 2i + j + k isa)30 degb)45 degc)60 degd)90 degCorrect answer is option 'C'. Can you explain this answer?
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The angle between two vectors a = i + 2j – k and b = 2i + j + k isa)30 degb)45 degc)60 degd)90 degCorrect answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The angle between two vectors a = i + 2j – k and b = 2i + j + k isa)30 degb)45 degc)60 degd)90 degCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The angle between two vectors a = i + 2j – k and b = 2i + j + k isa)30 degb)45 degc)60 degd)90 degCorrect answer is option 'C'. Can you explain this answer?.
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