Q4:

If $\vec{a} = \vec{b} + \vec{c}$, then is it true that $|\vec{a}| = |\vec{b}| + |\vec{c}|$? Justify your answer.

Step 1: Analyzing the Proposition

The question asks whether the magnitude of the sum of two vectors, $\vec{a} = \vec{b} + \vec{c}$, is necessarily equal to the sum of the magnitudes of the individual vectors, $|\vec{a}| = |\vec{b}| + |\vec{c}|$. [Per the Triangle Inequality for vectors, $|\vec{b} + \vec{c}| \leq |\vec{b}| + |\vec{c}|$.]

Step 2: Geometrical Interpretation

Geometrically, $\vec{a}$, $\vec{b}$, and $\vec{c}$ form the sides of a triangle (or a degenerate triangle). The condition $|\vec{b} + \vec{c}| = |\vec{b}| + |\vec{c}|$ holds if and only if vectors $\vec{b}$ and $\vec{c}$ are collinear and point in the same direction.

Step 3: Counter-Example Construction

To disprove the general statement, we choose non-collinear vectors $\vec{b}$ and $\vec{c}$.

Let $\vec{b} = \hat{i}$ and $\vec{c} = \hat{j}$.

Then $\vec{a} = \vec{b} + \vec{c} = \hat{i} + \hat{j}$.

Step 4: Magnitude Calculations

Calculate $|\vec{a}|$, $|\vec{b}|$, and $|\vec{c}|$:

• $|\vec{b}| = \sqrt{1^2 + 0^2} = 1$
• $|\vec{c}| = \sqrt{0^2 + 1^2} = 1$
• $|\vec{a}| = \sqrt{1^2 + 1^2} = \sqrt{2}$

Step 5: Verification of Inequality

Substitute these values into the expression $|\vec{b}| + |\vec{c}|$:

$|\vec{b}| + |\vec{c}| = 1 + 1 = 2$

Since $\sqrt{2} \approx 1.414$ and $1.414 \neq 2$, it is evident that:

$|\vec{a}| \neq |\vec{b}| + |\vec{c}|$

Step 6: Logical Justification

The equality $|\vec{a}| = |\vec{b}| + |\vec{c}|$ only holds when the angle $\theta$ between $\vec{b}$ and $\vec{c}$ is $0^\circ$. Given the definition of vector addition in $\mathbb{R}^3$, the magnitude is defined as:

$|\vec{b} + \vec{c}| = \sqrt{|\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{b}||\vec{c}|\cos\theta}$

For the equation $|\vec{a}| = |\vec{b}| + |\vec{c}|$ to be true, we require $\cos\theta = 1$, which implies $\theta = 0$. Since this is not true for all vectors, the statement is false in the general case.

Final Solution: No, it is not true in general. It is only true when $\vec{b}$ and $\vec{c}$ are collinear and in the same direction.