Q#8
Can a vector have zero component along a line and still have non-zero magnitude?Answer:
Yes. A non-zero magnitude vector has zero component along a line perpendicular to it.
Component along a line is cosθ times the vector, where θ is the angle between the line and vector's direction.
For θ = 90°, Cosθ = 0. So, the component along the line becomes zero.
Q#9
Let ε₁ and ε₂ be the angles made by vectors A and -A with the positive X-axis. Show that tan ε₁ = tan ε₂. Thus giving tan ε does not uniquely determine the direction of A.
Answer:
Since A and -A are opposite in direction ie angle between them is 180°. So, in this case, ε₁ = 180°+ ε₂. See Diagram below
So, tan ε₁ = tan (180°+ ε₂) = tan ε₂
Thus giving tan ε does not uniquely determine the direction of A because it indicates directions of both A and -A.
Q#10
Is the vector sum of the unit vectors i and j a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Answer:
Vector sum of the unit vectors i and j is not a unit vector because its magnitude is not unit. Yes, this sum can be multiplied by a scalar number to get a unit vector.
Vectors i and j represent vectors of unit magnitudes along X-axis and Y-axis, so the angle between them is a right angle. Hence the magnitude of their vector sum will be √2. If we multiply this sum by a scalar number 1/√2 we can get a unit vector.
Q#11
Let A = 3i + 4j. Write four vectors B such that A ≠ B but A = B.
Answer:
Four such vectors may be B = 3i – 4j, B = -3i + 4j, B = -3i – 4j and B = 5j
Q#12
Can you have A x B = A.B with A ≠0 and B ≠ 0? What if one of the two vectors is zero?
Answer:
If A ≠ 0 and B ≠ 0 then A x B cannot be equal to A.B because cross product is a vector quantity having a magnitude equal to ABsinθ and direction perpendicular to the plane containing A and B while the dot product A.B is a scalar quantity having its value equal to ABcosθ.
If one of the two vectors is zero then A.B = 0 and magnitude of A x B is also zero. In this case, the direction of the zero vector has no significance.
Q#13
If A x B = 0, can you say that (a) A = B ,(b) A ≠ B?
Answer:
If A x B = 0 then we can not say that (a) A = B or (b) A ≠ B. Since A x B = AB
Q#14
If A = 5i – 4j and B = -7.5i + 6j . Do we have B = kA? Can we say that B/A = k?
Answer:
In this case, we do have B = kA where k = -1.5. But we can not say that B/A = k because the division of vectors is not defined.
Component along a line is cosθ times the vector, where θ is the angle between the line and vector's direction.
For θ = 90°, Cosθ = 0. So, the component along the line becomes zero.
Q#9
Let ε₁ and ε₂ be the angles made by vectors A and -A with the positive X-axis. Show that tan ε₁ = tan ε₂. Thus giving tan ε does not uniquely determine the direction of A.
Answer:
Since A and -A are opposite in direction ie angle between them is 180°. So, in this case, ε₁ = 180°+ ε₂. See Diagram below
So, tan ε₁ = tan (180°+ ε₂) = tan ε₂
Thus giving tan ε does not uniquely determine the direction of A because it indicates directions of both A and -A.
Q#10
Is the vector sum of the unit vectors i and j a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Answer:
Vector sum of the unit vectors i and j is not a unit vector because its magnitude is not unit. Yes, this sum can be multiplied by a scalar number to get a unit vector.
Vectors i and j represent vectors of unit magnitudes along X-axis and Y-axis, so the angle between them is a right angle. Hence the magnitude of their vector sum will be √2. If we multiply this sum by a scalar number 1/√2 we can get a unit vector.
Q#11
Let A = 3i + 4j. Write four vectors B such that A ≠ B but A = B.
Answer:
Four such vectors may be B = 3i – 4j, B = -3i + 4j, B = -3i – 4j and B = 5j
Q#12
Can you have A x B = A.B with A ≠0 and B ≠ 0? What if one of the two vectors is zero?
Answer:
If A ≠ 0 and B ≠ 0 then A x B cannot be equal to A.B because cross product is a vector quantity having a magnitude equal to ABsinθ and direction perpendicular to the plane containing A and B while the dot product A.B is a scalar quantity having its value equal to ABcosθ.
If one of the two vectors is zero then A.B = 0 and magnitude of A x B is also zero. In this case, the direction of the zero vector has no significance.
Q#13
If A x B = 0, can you say that (a) A = B ,(b) A ≠ B?
Answer:
If A x B = 0 then we can not say that (a) A = B or (b) A ≠ B. Since A x B = AB
Q#14
If A = 5i – 4j and B = -7.5i + 6j . Do we have B = kA? Can we say that B/A = k?
Answer:
In this case, we do have B = kA where k = -1.5. But we can not say that B/A = k because the division of vectors is not defined.
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