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Two vectors vec(A) and vec(B) lie in pla...

Two vectors `vec(A) and vec(B)` lie in plane, another vector `vec(C )` lies outside this plane, then the resultant of these three vectors i.e., `vec(A)+vec(B)+vec(C )`

A

cannot be zero

B

can be zero

C

Lies in the plane of `vec(A)` or `vec(B)`

D

Lies in a plane different from that of any of the three vectors

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To solve the problem of determining the resultant of the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\), we will follow these steps: ### Step 1: Understand the Configuration of the Vectors - Vectors \(\vec{A}\) and \(\vec{B}\) lie in a plane, which we can denote as the XY-plane. - Vector \(\vec{C}\) lies outside this plane, which can be represented as being along the Z-axis (perpendicular to the XY-plane). **Hint:** Visualize the vectors in a 3D coordinate system to understand their orientation. ### Step 2: Analyze the Resultant of Vectors \(\vec{A}\) and \(\vec{B}\) - The resultant vector of \(\vec{A}\) and \(\vec{B}\) can be denoted as \(\vec{R} = \vec{A} + \vec{B}\). - Since both \(\vec{A}\) and \(\vec{B}\) lie in the same plane (XY-plane), the resultant \(\vec{R}\) will also lie in the same plane. **Hint:** Remember that the resultant of two vectors in the same plane will always remain in that plane. ### Step 3: Consider the Effect of Vector \(\vec{C}\) - Vector \(\vec{C}\) lies outside the plane of \(\vec{A}\) and \(\vec{B}\). This means that \(\vec{C}\) has a component that is perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\). - The resultant vector of \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) can be expressed as \(\vec{R_{total}} = \vec{R} + \vec{C}\). **Hint:** Think about how adding a vector that is perpendicular to the plane affects the overall direction of the resultant vector. ### Step 4: Determine if the Resultant Can Be Zero - Since \(\vec{R}\) lies in the XY-plane and \(\vec{C}\) lies along the Z-axis, they cannot cancel each other out. - The components of \(\vec{R}\) cannot counteract the component of \(\vec{C}\) that is perpendicular to the XY-plane. **Hint:** Recall that for two vectors to sum to zero, they must be equal in magnitude and opposite in direction. ### Step 5: Conclusion - Therefore, the resultant of the three vectors \(\vec{A} + \vec{B} + \vec{C}\) cannot be zero because \(\vec{C}\) introduces a component that is outside the plane of \(\vec{A}\) and \(\vec{B}\). - Thus, we conclude that the sum of these three vectors cannot be zero. **Final Answer:** The resultant of the vectors \(\vec{A} + \vec{B} + \vec{C}\) is not zero.
To solve the problem of determining the resultant of the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\), we will follow these steps: ### Step 1: Understand the Configuration of the Vectors - Vectors \(\vec{A}\) and \(\vec{B}\) lie in a plane, which we can denote as the XY-plane. - Vector \(\vec{C}\) lies outside this plane, which can be represented as being along the Z-axis (perpendicular to the XY-plane). **Hint:** Visualize the vectors in a 3D coordinate system to understand their orientation. ...
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