Q.The vector $\vec{a}+\vec{b}$ bisects the angle between the non-collinear vectors $\vec{a}$ and $\vec{b}$ if ________

The key idea is that a vector bisects an angle when it is equally inclined to the two given vectors — this happens when the vector is along the diagonal of a rhombus formed by equal-length vectors.

Step 1: For $\vec{a}+\vec{b}$ to bisect the angle between $\vec{a}$ and $\vec{b}$, the unit vectors along $\vec{a}$ and $\vec{b}$ must be symmetric about $\vec{a}+\vec{b}$. This requires $|\vec{a}| = |\vec{b}|$. …

Why This Works — The Geometry of Vector Addition

When you add two vectors $\vec{a}$ and $\vec{b}$, the resultant $\vec{a}+\vec{b}$ is the diagonal of the parallelogram formed by $\vec{a}$ and $\vec{b}$. The direction of this diagonal depends on both the directions and the lengths of $\vec{a}$ and $\vec{b}$.

Think of a rhombus — a parallelogram with all sides equal. In a rhombus, the diagonal does bisect the angle at the vertex. That’s the geometric intuition: if $\vec{a}$ and $\vec{b}$ have the same length, the parallelogram becomes a rhombus, and its diagonal (the sum) naturally bisects the angle between the sides.

If the lengths are different, the diagonal leans toward the longer vector, and the angle is split unevenly.

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Step-by-Step Reasoning

1. Set up the condition for angle bisection.

A vector $\vec{v}$ bisects the angle between $\vec{a}$ and $\vec{b}$ if it makes equal angles with both. That means the cosine of the angle between $\vec{v}$ and $\vec{a}$ equals the cosine of the angle between $\vec{v}$ and $\vec{b}$.

For $\vec{v} = \vec{a}+\vec{b}$, we require:

Since $|\vec{a}+\vec{b}| > 0$ (the vectors are non-collinear, so their sum is non-zero), we can cancel it from both denominators.

2. Expand the dot products.

Write $\vec{a}\cdot\vec{a} = |\vec{a}|^2$, $\vec{b}\cdot\vec{b} = |\vec{b}|^2$, and $\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$.

3. Simplify the equation.

Split each fraction:

Bring the dot product terms together:

Factor the right side:

4. Factor the difference of reciprocals.

Substitute:

Method: When Does a Vector Sum Bisect the Angle Between Two Vectors?

Use this reasoning pattern for any "does $\vec{a}+\vec{b}$ bisect the angle between $\vec{a}$ and $\vec{b}$" question.

Steps

Step 1: Recall the rhombus / unit-vector principle

The sum of two vectors is the diagonal of the parallelogram they span. A diagonal bisects the vertex angle only when that parallelogram is a rhombus — i.e. when the two spanning vectors have equal length.

Step 2: Translate "bisects" into equal angles

Require the resultant to make equal angles with each vector:

$\frac{(\vec{a}+\vec{b})\cdot\vec{a}}{|\vec{a}|}=\frac{(\vec{a}+\vec{b})\cdot\vec{b}}{|\vec{b}|},$ …

Common Mistakes

Mistake 1: Believing $\vec{a}+\vec{b}$ always bisects the angle

Why it's wrong: the diagonal of a general parallelogram leans toward the longer side, splitting the angle unevenly. Correct approach: it bisects only when the parallelogram is a rhombus, i.e. $|\vec{a}|=|\vec{b}|$.

Mistake 2: Confusing the condition with perpendicularity or with $\vec{a}=\vec{b}$

Why it's wrong: bisection needs equal lengths, not a right angle and not identical vectors. Correct approach: the single requirement is $|\vec{a}|=|\vec{b}|$; the directions may still differ. …