Questions: 若 A=10,B=5, 求以下各情形之 + 之量值及一之量值： （1） A / / B 且同向。 (2) A / / B 且反向。 （3） A 與 B 夾角 60° 。 （4） A 與 B 夾角 90° 。

Solution Steps

We are given the magnitudes of two vectors, \( |\vec{A}| = 10 \) and \( |\vec{B}| = 5 \). We need to find the magnitudes of their sum and difference in three different scenarios.

When \(\vec{A}\) and \(\vec{B}\) are parallel and in the same direction:

• The magnitude of the sum is \( |\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}| \).
• The magnitude of the difference is \( |\vec{A} - \vec{B}| = |\vec{A}| - |\vec{B}| \).

\[ |\vec{A} + \vec{B}| = 10 + 5 = 15 \] \[ |\vec{A} - \vec{B}| = 10 - 5 = 5 \]

\(\boxed{|\vec{A} + \vec{B}| = 15}\)

\(\boxed{|\vec{A} - \vec{B}| = 5}\)

When \(\vec{A}\) and \(\vec{B}\) are parallel and in opposite directions:

• The magnitude of the sum is \( |\vec{A} + \vec{B}| = ||\vec{A}| - |\vec{B}|| \).
• The magnitude of the difference is \( |\vec{A} - \vec{B}| = |\vec{A}| + |\vec{B}| \).

\[ |\vec{A} + \vec{B}| = |10 - 5| = 5 \] \[ |\vec{A} - \vec{B}| = 10 + 5 = 15 \]

\(\boxed{|\vec{A} + \vec{B}| = 5}\)

\(\boxed{|\vec{A} - \vec{B}| = 15}\)

When \(\vec{A}\) and \(\vec{B}\) form a \(60^\circ\) angle:

• The magnitude of the sum is given by the formula \( |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(60^\circ)} \).
• The magnitude of the difference is given by the formula \( |\vec{A} - \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(60^\circ)} \).

\[ |\vec{A} + \vec{B}| = \sqrt{10^2 + 5^2 + 2 \cdot 10 \cdot 5 \cdot \cos(60^\circ)} = \sqrt{100 + 25 + 50} = \sqrt{175} \approx 13.2288 \] \[ |\vec{A} - \vec{B}| = \sqrt{10^2 + 5^2 - 2 \cdot 10 \cdot 5 \cdot \cos(60^\circ)} = \sqrt{100 + 25 - 50} = \sqrt{75} \approx 8.6603 \]

\(\boxed{|\vec{A} + \vec{B}| \approx 13.2288}\)

\(\boxed{|\vec{A} - \vec{B}| \approx 8.6603}\)

Final Answer