3D geometry and vectors calculator
Type up to four points, vectors, lines and planes, and get everything ISC Mathematics asks about them: distances, angles, the foot of the perpendicular and the image of a point, the shortest distance between skew lines, where a line meets a plane, dot and cross products, areas and the plane through three points - exact answers, with surds.
Read as
- a vector 2i - j + 3k
- b vector i + j + k
a and b
- a · b
- 4
- a × b
- -4i + j + 3k
- Area of the parallelogram |a × b|
- √26 ≈ 5.099
- Area of the triangle ½|a × b|
- √26/2 ≈ 2.55
- Angle between them
- cos⁻¹(2√42/21) rad ≈ 51.89°
- Projection of a on b
- 4√3/3 ≈ 2.309
- Relation
- neither parallel nor perpendicular
a (vector)
- Magnitude
- √14 ≈ 3.742
- Unit vector
- (2i - j + 3k) / √14
- Direction cosines
- √14/7, -√14/14, 3√14/14
b (vector)
- Magnitude
- √3 ≈ 1.732
- Unit vector
- (i + j + k) / √3
- Direction cosines
- √3/3, √3/3, √3/3
Practise on real ISC questions
- If $\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = 5\hat{i} - 3\hat{j} + \hat{k}$, find the… 2024
- Find the equation of the plane passing through the point $(2, -1, 3)$ and containing the line… 2025 Practice
- Consider the following statements and choose the correct option: Statement 1: If $\vec{a}$ and $\vec{b}$… 2026 Specimen
- If $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$ and $\vec{a} + \vec{b} + \vec{c} = 0$, then verify… 2025 Practice
- If $\vec{a}$ and $\vec{b}$ are unit vectors enclosing an angle $\theta$ and $|\vec{a}+\vec{b}| < 1$, then… 2026 Specimen
- The angle between the two planes $x + y + 2z = 9$ and $2x - y + z = 15$ is: 2025
- Show that… 2018 Specimen
- The angle between the line $\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{2}$ and the plane… 2025 Practice
- A scalar is multiplied by a unit vector, then the resultant is: Statement 1: A vector with the magnitude of… 2026
- Choose the correct option for the following questions. The equation of the plane which is parallel to… 2022 Specimen
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