Table of Contents
What is a dot B cross C?
The scalar triple product of three vectors a, b, and c is (a×b)⋅c. It is a scalar product because, just like the dot product, it evaluates to a single number. (In this way, it is unlike the cross product, which is a vector.)
Is AXB xC same as ax BxC?
That is, a x (b x c) lies in the plane of b and c. Consequently, (a x b) x c, which is the same as −c x (a x b), will lie in the plane of a and b. Hence, (a x b) x c will, in general, be different from a x (b x c).
What is the projection of vector A on vector B?
The vector projection of a vector on a vector other than zero b (also known as vector component or vector resolution of a in the direction of b) is the orthogonal projection of a on a straight line parallel to b. It is a parallel vector a b, defined as the scalar projection of a on b in the direction of b.
When the scalar triple product of three vectors vanishes then the vectors are?
If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume. This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.
What is the dot product of a vector with itself?
A · B = B · A. Since the projection of a vector on to itself leaves its magnitude unchanged, the dot product of any vector with itself is the square of that vector’s magnitude. A · A = AA cos 0° = A 2. Applying this corollary to the unit vectors means that the dot product of any unit vector with itself is one.
What is the vector 1 3 V = (2 3)?
Thus, the vector 1 3 v = ( 2 3, 1 3, − 2 3) is a unit vector in the same direction as v . In general, for v ≠ 0, we can scale (or normalize) v to the unit vector v ‖ v ‖ pointing in the same direction as v. Let u = ( u 1, u 2, u 3) and v = ( v 1, v 2, v 3).
What happens when you multiply a vector by a scalar?
Multiplication of a vector by a scalar changes the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar “scales” the vector. Multiplication of a vector by a scalar is distributive.
How do you find the direction of a unit vector?
The vector n̂ (n hat) is a unit vector perpendicular to the plane formed by the two vectors. The direction of n̂ is determined by the right hand rule, which will be discussed shortly. The cross product is distributive… A × (B + C) = (A × B) + (A × C)