Vector Addition: The Tip-to-Tail Method

Adding two vectors geometrically is one of the simplest and most beautiful operations in physics: place the tail of B at the tip of A, and the resultant goes from the tail of A to the tip of B. The component-wise addition is just the algebraic shadow of this geometric picture.

The Method

For vectors $\vec A = (A_x, A_y)$ and $\vec B = (B_x, B_y)$:

$\vec A + \vec B = (A_x + B_x,\, A_y + B_y)$

The magnitude of the resultant is:

$|\vec A + \vec B| = \sqrt{(A_x + B_x)^{2} + (A_y + B_y)^{2}}$

And the direction (measured from the positive $x$-axis) is:

$\theta = \arctan\!\left(\dfrac{A_y + B_y}{A_x + B_x}\right)$

Step-by-Step Solution

Given: $\vec A = (3, 4)$ and $\vec B = (2, -1)$.

Find: $\vec A + \vec B$, its magnitude, and its direction.

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Step 1 — Add the components.

$(\vec A + \vec B)_x = A_x + B_x = 3 + 2 = 5$

$(\vec A + \vec B)_y = A_y + B_y = 4 + (-1) = 3$

So:

$\vec A + \vec B = (5, 3)$

Step 2 — Compute the magnitude using the Pythagorean theorem.

$|\vec A + \vec B| = \sqrt{5^{2} + 3^{2}} = \sqrt{25 + 9} = \sqrt{34} \approx 5.831$

Step 3 — Compute the direction (angle from positive $x$-axis).

$\theta = \arctan\!\left(\dfrac{3}{5}\right) = \arctan(0.6) \approx 30.96°$

Step 4 — Sanity check by computing the magnitudes of $\vec A$ and $\vec B$.

$|\vec A| = \sqrt{9 + 16} = \sqrt{25} = 5.000$

$|\vec B| = \sqrt{4 + 1} = \sqrt{5} \approx 2.236$

Notice $|\vec A + \vec B| \approx 5.831$, which is less than $|\vec A| + |\vec B| \approx 7.236$. The triangle inequality says equality only holds when $\vec A$ and $\vec B$ point in exactly the same direction. Here they don't, so the resultant is shorter than the sum of magnitudes.

Step 5 — Verify with the law of cosines.

The angle between $\vec A$ and the negative of $\vec B$ (which closes the triangle) is the angle of the parallelogram's interior angle. With a bit of algebra:

$|\vec A + \vec B|^{2} = |\vec A|^{2} + |\vec B|^{2} + 2\vec A\cdot\vec B$

Compute the dot product: $\vec A\cdot\vec B = 3(2) + 4(-1) = 6 - 4 = 2$.

$|\vec A + \vec B|^{2} = 25 + 5 + 2(2) = 34$

$|\vec A + \vec B| = \sqrt{34} \approx 5.831 \;\;\checkmark$

Both methods agree.

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Answer:

$\boxed{\;\vec A + \vec B = (5, 3),\quad |\vec A + \vec B| = \sqrt{34} \approx 5.831,\quad \theta \approx 30.96°\;}$

Visually, the vector $\vec A$ is drawn from the origin pointing up-and-right, $\vec B$ is then drawn starting at the tip of $\vec A$ pointing slightly down-and-right, and the resultant $\vec A + \vec B$ closes the triangle from the origin to the tip of $\vec B$.

Try It

• Adjust the components of A and B with the sliders.
• The visualization redraws the tip-to-tail diagram and the resultant in real time.
• The HUD shows the component sum, magnitude, and direction.
• Notice that swapping the order of addition (B then A) gives the same resultant — vectors commute!