Answer :
Let's analyze the given options regarding vector properties and their resultant vectors.
1) Two equal vectors can never give zero resultant:
- This statement is incorrect. Two vectors that have the same magnitude and direction will sum to a vector that is twice their magnitude and in the same direction. However, if these vectors are equal in magnitude but opposite in direction, their resultant can indeed be zero. Therefore, this statement is false.
2) Three non-coplanar vectors cannot give zero resultant:
- Non-coplanar vectors are vectors that do not lie in the same plane. For three vectors to sum to zero, they must be able to form a closed triangle when placed head-to-tail. However, for vectors to be non-coplanar, it means they do not lie in the same plane, hence it's impossible for them to form a closed triangle. Therefore, this statement is true.
3) If \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\) and \(|\vec{a}| \neq |\vec{b}| \neq |\vec{c}|\), then \(\vec{a} + \vec{b} + \vec{c}\) can never be a null vector:
- The given condition \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\) implies that vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar. However, given that their magnitudes are distinct, it does not necessarily follow that their vector sum must be zero. This is a more nuanced condition and usually does not strictly lead to the vector sum being zero. Thus, this statement is false.
4) If \(\vec{a} \times \vec{b}=0\) and \(|\vec{a}|=|\vec{b}|\), then \(\vec{a}+\vec{b}\) can be zero:
- The condition \(\vec{a} \times \vec{b} = 0\) implies that vectors \(\vec{a}\) and \(\vec{b}\) are parallel or one of them is a zero vector. Since \(|\vec{a}| = |\vec{b}|\), it indicates that the vectors have the same magnitude. If these vectors are in the opposite direction, their vector sum will indeed be zero. Thus, this statement is correct.
Therefore, the correct statement is:
Option 4: If [tex]\(\vec{a} \times \vec{b}=0\)[/tex] and [tex]\(|\vec{a}|=|\vec{b}|\)[/tex], then [tex]\(\vec{a}+\vec{b}\)[/tex] can be zero.
1) Two equal vectors can never give zero resultant:
- This statement is incorrect. Two vectors that have the same magnitude and direction will sum to a vector that is twice their magnitude and in the same direction. However, if these vectors are equal in magnitude but opposite in direction, their resultant can indeed be zero. Therefore, this statement is false.
2) Three non-coplanar vectors cannot give zero resultant:
- Non-coplanar vectors are vectors that do not lie in the same plane. For three vectors to sum to zero, they must be able to form a closed triangle when placed head-to-tail. However, for vectors to be non-coplanar, it means they do not lie in the same plane, hence it's impossible for them to form a closed triangle. Therefore, this statement is true.
3) If \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\) and \(|\vec{a}| \neq |\vec{b}| \neq |\vec{c}|\), then \(\vec{a} + \vec{b} + \vec{c}\) can never be a null vector:
- The given condition \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\) implies that vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar. However, given that their magnitudes are distinct, it does not necessarily follow that their vector sum must be zero. This is a more nuanced condition and usually does not strictly lead to the vector sum being zero. Thus, this statement is false.
4) If \(\vec{a} \times \vec{b}=0\) and \(|\vec{a}|=|\vec{b}|\), then \(\vec{a}+\vec{b}\) can be zero:
- The condition \(\vec{a} \times \vec{b} = 0\) implies that vectors \(\vec{a}\) and \(\vec{b}\) are parallel or one of them is a zero vector. Since \(|\vec{a}| = |\vec{b}|\), it indicates that the vectors have the same magnitude. If these vectors are in the opposite direction, their vector sum will indeed be zero. Thus, this statement is correct.
Therefore, the correct statement is:
Option 4: If [tex]\(\vec{a} \times \vec{b}=0\)[/tex] and [tex]\(|\vec{a}|=|\vec{b}|\)[/tex], then [tex]\(\vec{a}+\vec{b}\)[/tex] can be zero.
