Vectors

How do you add two vectors?

Quick answer

You add two vectors geometrically by placing the tail of the second at the head of the first, drawing the resultant vector from the initial tail to the final head. Algebraically, add their matching horizontal and vertical components. For vectors u = (x1, y1) and v = (x2, y2), the sum is (x1 + x2, y1 + y2).

The topic explained

Vectors, scalars, and the basic idea of addition

A scalar has only size, like a mass of 5 kilograms or a speed of 30 miles per hour. A vector has both size, known as magnitude, and a direction. Displacement, velocity, and force are vectors. Two vectors are equal only when they share the exact same magnitude and the exact same direction, regardless of where they start on a coordinate plane.

Adding two vectors combines their independent shifts to find a single net result, called the resultant vector. If you walk 4 meters east and then 3 meters north, your total displacement is not simply 7 meters. It is a straight line pointing northeast from your starting point.

The geometric methods: triangle and parallelogram rules

In the triangle rule, you draw the first vector as an arrow. Next, attach the tail of the second vector to the tip, or head, of the first vector. The resultant vector is the arrow drawn from the very start of the first vector to the tip of the second vector.

In the parallelogram rule, you place the tails of both vectors at the same starting point. Complete a parallelogram using those two vectors as adjacent sides. The resultant vector is the diagonal that begins at the common tail and reaches across to the opposite corner. Both geometric rules yield identical directions and lengths.

The algebraic method: adding components and finding length

When working on a Cartesian plane, vectors are written in component form as pairs of horizontal and vertical changes. If vector u = (x1, y1) and vector v = (x2, y2), their sum is u + v = (x1 + x2, y1 + y2). For example, adding u = (4, -2) and v = (-1, 6) gives (4 + (-1), -2 + 6) = (3, 4).

To find the length or magnitude of this resultant vector, use the Pythagorean theorem: the square root of (x-squared plus y-squared). For the vector (3, 4), the magnitude is the square root of (9 + 16), which is the square root of 25, or 5 units.

Free printable worksheet

Adding and Subtracting Vectors in the Plane

This practice sheet is for 12th-grade students studying introductory vectors. Solve each problem by finding the components or magnitude as requested.

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Personal AI lessons and practice, from preschool to university

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Adding and Subtracting Vectors in the PlaneName:

Identify whether a wind blowing at 15 knots from the northwest is a scalar or a vector.
Given vector a = (3, 5) and vector b = (2, 1), find the component form of vector a + vector b.
Given vector u = (-4, 7) and vector v = (6, -3), calculate the sum u + v.
Find the magnitude of the vector v = (6, 8).
Vector p has components (5, -2). Write the components of the opposite vector, -p.
Subtract vector w = (3, 9) from vector z = (7, 4). Calculate z - w.
A plane flies due east for 120 miles, then turns due north for 50 miles. What is the total straight-line displacement from the start?
Vectors m = (1, k) and n = (4, -2) add up to a resultant vector with a magnitude of 5 along the positive x-axis, so m + n = (5, 0). Find the value of k.

Common questions

What is the difference between vector addition and regular addition?

Regular addition combines single numerical quantities without direction. Vector addition accounts for both magnitude and direction, meaning 3 units plus 4 units can equal anything between 1 unit and 7 units, depending on the angle between them.

How do you subtract one vector from another geometrically?

To subtract vector v from vector u, flip vector v so it points in the opposite direction to create -v. Then, add u and -v using the standard triangle rule: attach the tail of -v to the head of u.

What are the most common mistakes students make when adding vectors?

A frequent mistake is adding the magnitudes directly rather than combining components. Another common error is connecting vectors head-to-head instead of placing the tail of the second vector at the head of the first.

What topic comes right after vector addition in 12th grade?

Students typically study scalar multiplication next, followed by linear combinations and the dot product to find angles between vectors.

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