Vectors and the Geometry of Space: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 1, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A quantity carrying both magnitude and direction, represented by an arrow or by an ordered triple of components; two arrows represent the same vector when one is a parallel translate of the other.
Position vector
The vector from the origin to a point P, whose components are exactly the coordinates of P. It is the bridge that lets geometric statements about points become algebra about vectors.
Vector magnitude
The length of a vector, computed as the square root of the sum of the squares of its components. It is zero only for the zero vector, and scaling a vector by c multiplies the magnitude by the absolute value of c.
Unit vector
A vector of length one, obtained by dividing any nonzero vector by its own magnitude. Unit vectors carry direction only, which is why they appear in direction cosines and directional derivatives.
Standard basis vectors i, j, k
The three mutually perpendicular unit vectors along the positive x, y and z axes. Every vector in space is a unique linear combination of them, which is what makes component arithmetic legitimate.
Dot product
The scalar formed by summing the products of matching components, equal also to the product of the two magnitudes with the cosine of the angle between them. It is commutative and distributes over addition.
Orthogonality test
Two nonzero vectors are perpendicular exactly when their dot product is zero. A dot product of zero involving a zero vector says nothing about angle, since the zero vector has no direction.
Scalar projection
The signed length of one vector's shadow on another, computed as the dot product divided by the magnitude of the vector being projected onto. The sign records whether the shadow points with or against that direction.
Vector projection
The scalar projection times the unit vector in the target direction, giving the component of one vector lying along another. Subtracting it leaves the orthogonal component, the basis of decomposing forces.
Cross product
A vector defined for two vectors in space, perpendicular to both, with magnitude equal to the product of lengths times the sine of the included angle. It is anticommutative, so swapping the inputs reverses the result.
Right-hand rule
The orientation convention fixing the direction of a cross product: curl the right hand from the first vector toward the second and the thumb gives the product direction. Reversing the order flips the thumb.
Parallel test via cross product
Two nonzero vectors are parallel exactly when their cross product is the zero vector, since the sine of the angle vanishes only at zero or a straight angle.
Scalar triple product
The dot of one vector with the cross product of two others, equal to a three-by-three determinant. Its absolute value is the volume of the parallelepiped spanned, and a zero value means the three vectors are coplanar.
Area of a parallelogram in space
The magnitude of the cross product of the two edge vectors. Half that number gives the area of the triangle they determine, a fact used constantly in surface integrals.
Vector equation of a line
A point on the line plus a scalar parameter times a direction vector. The same line has infinitely many such equations, since the base point and the scale of the direction may both change.
Parametric equations of a line
The three component equations obtained by writing out a vector line equation coordinate by coordinate, each coordinate an affine function of the parameter.
Symmetric equations of a line
The form obtained by solving each parametric equation for the parameter and setting the results equal. They break down whenever a direction component is zero, in which case that coordinate is held constant instead.
Skew lines
Two lines in space that neither intersect nor run parallel. They exist only in three or more dimensions, and the distance between them is measured along their common perpendicular direction.
Normal vector to a plane
A nonzero vector perpendicular to every vector lying in the plane. It determines the plane's tilt completely, so a plane is fixed by one point together with one normal.
Scalar equation of a plane
The statement that the dot product of a normal vector with the vector from a fixed point to a general point is zero, which expands into a linear equation in x, y and z.
Angle between two planes
The angle between their normal vectors, taken as the acute value. Two planes are parallel when the normals are parallel and perpendicular when the normals have zero dot product.
Distance from a point to a plane
The absolute scalar projection of a vector from any point of the plane to the given point onto the unit normal. Choosing a different plane point does not change the answer.
Quadric surface
A surface in space defined by a second-degree equation in the three coordinates. Ellipsoids, paraboloids, cones and the two hyperboloids are the standard types, distinguished by their signs.
Trace of a surface
The curve of intersection between a surface and a plane parallel to a coordinate plane. Sketching several traces is the practical way to identify and draw a quadric by hand.
Cylinder in three dimensions
A surface generated by moving a line parallel to a fixed direction along a plane curve. In equation form it is recognised by a missing variable, which means that variable is unconstrained.