Question

layla claims that the magnitude of the sum of two vectors is never equivalent to the sum of the magnitudes of the two vectors. which diagram provides a counterexample to laylas claim?

Explanation:

Step1: Recall vector addition rule

When two vectors \(\vec{v}\) and \(\vec{w}\) are in the same direction (collinear and same sense), the magnitude of their sum \(|\vec{v}+\vec{w}| = |\vec{v}|+|\vec{w}|\). This is because we can use the triangle law or the parallelogram law of vector addition. If they are in the same direction, the resultant vector's magnitude is the sum of their individual magnitudes.

Step2: Analyze each diagram

• First diagram: Vectors \(\vec{v}\) and \(\vec{w}\) are in the same direction (both pointing to the right). So when we add them \(\vec{v}+\vec{w}\), the magnitude of the resultant vector should be \(|\vec{v}| + |\vec{w}|\), which would be a counterexample to Layla's claim (since she said it's never equivalent).
• Second diagram: Vectors \(\vec{v}\) and \(\vec{w}\) are not in the same direction (one is going up - right, the other down - right). The resultant \(\vec{v}+\vec{w}\) will have a magnitude less than \(|\vec{v}|+|\vec{w}|\) (by triangle inequality \(|\vec{v}+\vec{w}|\leq |\vec{v}| + |\vec{w}|\), and equality holds only when they are in the same direction).
• Third diagram: Vectors \(\vec{v}\) and \(\vec{w}\) are not in the same direction (one up - right, one down - left). The resultant magnitude will be less than \(|\vec{v}|+|\vec{w}|\).
• Fourth diagram: Vectors \(\vec{v}\) and \(\vec{w}\) are in opposite directions. The magnitude of the resultant will be \(||\vec{v}|-|\vec{w}||\), which is less than \(|\vec{v}|+|\vec{w}|\) (assuming \(|\vec{v}|

eq |\vec{w}|\)) or zero (if \(|\vec{v}| = |\vec{w}|\)), but not equal to the sum.

Answer:

The first diagram (with vectors \(\vec{v}\), \(\vec{w}\) in the same direction and \(\vec{v}+\vec{w}\) as their sum in the same direction) provides the counterexample.