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° 。

Transcript text: 若 $|\mathrm{A}|=10,|\mathrm{~B}|=5$, 求以下各情形之 + 之量值及一之量值：（1） $\vec{A} / / \vec{B}$ 且同向。 (2) $\overrightarrow{\mathrm{A}} / / \overrightarrow{\mathrm{B}}$ 且反向。（3） $\overrightarrow{\mathrm{A}}$ 與 $\overrightarrow{\mathrm{B}}$ 夾角 $60^{\circ}$ 。（4） $\overrightarrow{\mathrm{A}}$ 與 $\overrightarrow{\mathrm{B}}$ 夾角 $90^{\circ}$ 。

Solution

Solution Steps

Step 1: Understanding the Problem

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.

Step 2: Vectors are Parallel and in the Same Direction

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}| $.

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

$\boxed{|\vec{A} + \vec{B}| = 15}$

$\boxed{|\vec{A} - \vec{B}| = 5}$

Step 3: Vectors are Parallel and in Opposite Directions

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}| $.

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

$\boxed{|\vec{A} + \vec{B}| = 5}$

$\boxed{|\vec{A} - \vec{B}| = 15}$

Step 4: Vectors Form a 60-Degree Angle

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

When $\vec{A}$ and $\vec{B}$ are parallel and in the same direction:
- $\boxed{|\vec{A} + \vec{B}| = 15}$
- $\boxed{|\vec{A} - \vec{B}| = 5}$
When $\vec{A}$ and $\vec{B}$ are parallel and in opposite directions:
- $\boxed{|\vec{A} + \vec{B}| = 5}$
- $\boxed{|\vec{A} - \vec{B}| = 15}$
When $\vec{A}$ and $\vec{B}$ form a $60^\circ$ angle:
- $\boxed{|\vec{A} + \vec{B}| \approx 13.2288}$
- $\boxed{|\vec{A} - \vec{B}| \approx 8.6603}$