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Question:
Given a + b + c + d = 0, which of the following statements are correct :

(a) a, b, c, and d must each be a null vector,
(b) The magnitude of (a + c) equals the magnitude of ( b + d),

(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear?

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Last updated date: 13th Jun 2024
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Answer
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Hint: You can use vector properties and algebra to determine the correctness of these statements.


Step-by-Step Solutions:

(a) a, b, c, and d must each be a null vector:

Answer: False.

Explanation: The equation a + b + c + d = 0 does not imply that each vector (a, b, c, d) must be a null vector. It means that the sum of these vectors results in a null vector, but the individual vectors can still have non-zero magnitudes.


(b) The magnitude of (a + c) equals the magnitude of (b + d):

Answer: True.

Explanation: This statement is true. You can prove it using the properties of vector addition and subtraction. If a + b + c + d = 0, then (a + c) = -(b + d), which means that the magnitudes of (a + c) and (b + d) are equal.


(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d:

Answer: False.

Explanation: This statement is not necessarily true. The equation a + b + c + d = 0 does not impose any restriction on the magnitude of individual vectors. It's possible for the magnitude of 'a' to be greater than the sum of the magnitudes of 'b,' 'c,' and 'd.'


(d) b + c must lie in the plane of a and d if a and d are not collinear and in the line of a and d if they are collinear:

Answer: True.

Explanation: This statement is true. If vectors 'a' and 'd' are not collinear (i.e., they do not lie on the same line), then the vector sum 'b + c' will lie in the plane defined by 'a' and 'd.' If 'a' and 'd' are collinear (i.e., they lie on the same line), then 'b + c' will lie along the same line as 'a' and 'd.'


Note: These statements can be understood by applying the properties of vector addition and algebra. The key is to remember that the equation a + b + c + d = 0 represents a balance of vector quantities, and it doesn't impose specific conditions on the magnitudes of the individual vectors.

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