Vector Calculator

Vectors are mathematical objects possessing both magnitude (length) and direction. In a two-dimensional Cartesian coordinate system, a vector \(\mathbf{a}\) is defined by components \((a_1, a_2)\), while three-dimensional vectors utilize \((a_1, a_2, a_3)\). The magnitude of a vector is calculated using the Pythagorean theorem generalization: \(|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\). To find the dot product between two vectors \(\mathbf{a}\) and \(\mathbf{b}\), you multiply corresponding components and sum the results: \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\). Furthermore, the cross product in 3D yields a new vector orthogonal to the original pair, computed via determinant expansion of standard unit vectors. Vector projections determine how much of one vector travels in the direction of another, utilizing the dot product divided by the squared magnitude of the target vector.

Let us walk through calculating the dot product, angle, and magnitude for two 3D vectors. Suppose vector \(\mathbf{a} = (1, 2, 3)\) and vector \(\mathbf{b} = (4, 5, 6)\). First, we calculate the magnitude of vector \(\mathbf{a}\): \(|\mathbf{a}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \approx 3.74\). Next, we find the dot product \(\mathbf{a} \cdot \mathbf{b} = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32\). To find the angle \(\theta\) between them, we use the formula \(\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\). The magnitude of \(\mathbf{b}\) is \(\sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77} \approx 8.77\). Thus, \(\cos(\theta) = \frac{32}{3.74 \times 8.77} \approx 0.975\), giving an angle \(\theta \approx 12.93^\circ\).

When working with vectors, always double-check your dimension settings (2D vs. 3D) before entering coordinates to avoid calculation errors. Remember that the dot product yields a scalar (a single number), whereas the cross product yields an orthogonal vector. When calculating vector projections, ensure you are projecting the correct vector onto the target vector, as the operation is non-commutative.