Example Question - parallel vectors
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Analyzing Vectors and Angles
بما أن المتجهات متوازية ومتعاكسة الاتجاه على نفس الخط، يمكننا استخدام مبدأ المتجهات المتعادلة ونضع معادلة للمتجهات: \[ 3t + 4t = 7t \] \[ 7t = 7t \] نلاحظ أن المعادلة صحيحة، وبالتالي القيمة \( t \) هي قيمة مقبولة. لذا اتجاه الموجات على الخط المستقيم الذي يحتوي على المتجهين \( 3t \) و \( 7t \) يكون صحيحًا.
Identifying Parallel Vectors with Scalar Multiples
Vectors are considered to be parallel if they are scalar multiples of each other. In other words, if you can multiply one vector by a scalar (a number) and get the other vector, the two vectors are parallel. The reference vector given is (2, 10). To find parallel vectors, we look for the vectors whose components can be obtained by multiplying the reference vector's components by the same scalar. Starting with the reference vector (2, 10), let's check each one against this: 1. (-2, 10): Multiplying the reference vector (2, 10) by -1 gives us (-2, -10), not (-2, 10), so this vector is not parallel. 2. (-2, -10): This vector can be obtained by multiplying the reference vector (2, 10) by the scalar -1, so this vector is parallel. 3. (4, 20): This vector can be obtained by multiplying the reference vector (2, 10) by the scalar 2, so this vector is parallel. 4. (4, 5): There's no single scalar that can multiply (2, 10) to get (4, 5); scaling the first component by 2 would require scaling the second component by 0.5, so this vector is not parallel. 5. (1, 5): Multiplying (2, 10) by 0.5 would indeed give us (1, 5), so this vector is parallel. 6. (10, 2): No scalar multiplication of (2, 10) will result in (10, 2), so this vector is not parallel. 7. (3, 11): No scalar multiplication of (2, 10) will result in (3, 11), so this vector is not parallel. The vectors parallel to (2, 10) are: (-2, -10), (4, 20), (1, 5)
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