Free Vector Projection Calculator

Understanding Vector Projection

The orthogonal projection of one vector onto another reveals how much of the first vector contributes to the direction of the second. Picture two vectors a and b in space that do not lie on the same line. When a light is cast from above, the silhouette that a forms along the line of b is exactly the orthogonal projection (commonly denoted proj). This intuitive “shadow” interpretation is why the operation is often called the projection of vector a onto b.

The Vector Projection Formula

The mathematical expression for calculating this projection uses the dot product:

$$\operatorname{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \,\mathbf{b}$$

Here, $\mathbf{a} \cdot \mathbf{b}$ is the dot product (also called the scalar product) of the two vectors. The denominator $\mathbf{b} \cdot \mathbf{b}$ equals $\|\mathbf{b}\|^2$, the squared length of b. The result is a vector that lies on b and whose magnitude is the scalar projection (or component) of a onto b.

The scalar projection itself is given by:

$$\operatorname{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|}$$

Thus, the vector projection equals the scalar projection multiplied by the unit vector in the direction of b. Because the formula relies on the dot product—which can be computed with a dedicated dot product calculator—the same expression works for vectors of any dimension, though practical use usually involves two or three components.

Derivation of the Formula

To understand where the formula comes from, decompose a into two orthogonal parts: one parallel to b (the projection) and one perpendicular to b (the rejection). Because the projection is parallel to b, it can be written as $C\,\mathbf{b}$ for some scalar $C$. The rejection vector ort satisfies $\mathbf{a} = C\mathbf{b} + \mathbf{ort}$. Take the dot product of both sides with b:

$$\mathbf{a}\cdot\mathbf{b} = C(\mathbf{b}\cdot\mathbf{b}) + \mathbf{ort}\cdot\mathbf{b}$$

Since ort is orthogonal to b, their dot product is zero, giving $C = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}$. Substituting back yields the vector projection formula. This derivation highlights why the result is also called the orthogonal projection formula, and why an orthogonal projection calculator follows the same procedure.

Using the Orthogonal Projection Calculator

Consider the three‑dimensional vectors a = [2, –3, 5] and b = [3, 6, –4]. Using an online vector projection calculator (or completing the steps manually) gives:

• Dot product: $\mathbf{a}\cdot\mathbf{b} = 2\times3 + (-3)\times6 + 5\times(-4) = -32$
• Norm squared of b: $\mathbf{b}\cdot\mathbf{b} = 3^2 + 6^2 + (-4)^2 = 61$
• Coefficient: $C = -32 / 61 \approx -0.5246$
• Projection vector: $C \,\mathbf{b} = \dfrac{-32}{61} [3, 6, -4] = \left[-\dfrac{96}{61}, -\dfrac{192}{61}, \dfrac{128}{61}\right] \approx [-1.5738, -3.1475, 2.0984]$

This result matches the output produced by the tool.

Special Case: Orthogonal Vectors

If a and b are perpendicular, their dot product is zero. Substituting this into the projection formula gives the zero vector, meaning a has no component along b. For example, a = [2, 6, –3] and b = [6, 4, 12] have a dot product of zero, so the projection is [0, 0, 0]. The orthogonal projection calculator correctly returns this null result, confirming the geometric intuition that an orthogonal vector casts no shadow.

Physical Application: Forces on an Inclined Plane

Vector projection appears naturally in physics when decomposing forces. Imagine a cart on a hill. The gravitational force F = [0, –400] N acts vertically downward. To find the force component pulling the cart along the slope, project F onto a direction vector parallel to the hill. With a slope angle $\alpha = 45^\circ$, a convenient unit direction vector is u = [cos45°, sin45°] ≈ [0.7071, 0.7071].

Because u is a unit vector ($\mathbf{u}\cdot\mathbf{u}=1$), the projection simplifies to:

$$\operatorname{proj} = (\mathbf{F}\cdot\mathbf{u})\,\mathbf{u}$$

Compute $\mathbf{F}\cdot\mathbf{u} = 0\cdot0.7071 + (-400)\cdot0.7071 = -282.84$. Thus, the projection vector is –282.84 u ≈ [–200, –200]. Its magnitude, 282.84 N, is the force that must be counteracted to keep the cart stationary. Steeper slopes (larger α) increase this value, while shallower slopes reduce it. This example illustrates how the vector projection formula directly translates into a measurable physical quantity.

Role in Data Science

Vector projection is a key operation in linear regression and principal component analysis (PCA). These methods project high‑dimensional data onto lower‑dimensional subspaces while preserving as much variance as possible. The projection extracts the directions that best represent the data, making it a cornerstone of many machine‑learning algorithms. Whether you are working on mechanics, data analysis, or linear algebra, the orthogonal projection calculator (also called a vector projection calculator) provides a quick and reliable way to compute the projection of a onto b using the vector projection formula and dot product operations.