Mathematics · Physics

Vector Algebra and Calculus

192 Questions

Vector algebra involves mathematical operations on spatial quantities including dot products and cross products. These questions test the understanding of vector spaces and linear combinations. This topic is crucial for advanced mathematics and physics exams.

Dot and cross productsVector linear combinationsPerpendicular vector calculationsVector space dimensionsCollinear points and vectors

Vector Algebra and Calculus Questions

Multiple choice technology programming languages

Output of: vector (1 2 2 3 4 4)

  1. #6(1 2 2 3 4)

  2. #6(1 2 3 4)

  3. #6(1 2 2 3 4 4)

  4. #4(1 2 3 4)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In Scheme, vectors created from lists remove duplicates from the list structure. (1 2 2 3 4 4) has 6 elements but only 5 unique values. Scheme vector notation #6(...) shows length 6 but deduplicates values, resulting in #6(1 2 2 3 4).

Multiple choice

If a vector field $\vec V$is related to another vector field $\vec A$through $\vec V$= $\nabla$x $\vec A$, which of the following is true? (Note : C and Sc refer to any closed contour and any surface whose boundary is C.)
  1. $\oint_0 \vec V. \vec{dl} = \int_z \int \vec A. \vec d S$
  2. $\oint_0 \vec A. \vec{dl} = \int_z \int \vec V. \vec d S$
  3. $\oint_0 \vec V. \vec{dl} = \int_z \int \Delta x \vec A. \vec d S$
  4. $\oint_0 \Delta x \vec V. \vec{dl} = \int_z \int \vec V. \vec d S$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiple choice

$\nabla$ × P, where P is a vector, is equal to
  1. Px $\nabla$ x P − $\nabla^2$P
  2. $\nabla^2$P + $\nabla$($\nabla$.P)
  3. $\nabla^2$P + $\nabla$x P
  4. $\nabla$($\nabla$.P) - $\nabla^2$P
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\text{The vector Triple product is} \\ A \times (B \times C) = B(A.C) - C(A.B) \\ Thus \hspace{0.2cm} \nabla \times \nabla \times P = \nabla(\nabla .P) P(\nabla.\nabla) = \nabla(\nabla . P) - \nabla^2 P$

Multiple choice

It is given that X1, X2, …..XM are M non-zero orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X1, X2, …XM , - XM is

  1. 2M

  2. M + 1

  3. M

  4. dependent on the choice of X1, X2, …..XM
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If X_1, X_2, ..., X_M are orthogonal non-zero vectors, they are linearly independent and span an M-dimensional subspace. The set {-X_1, -X_2, ..., -X_M} consists of the same vectors scaled by -1, which doesn't add new dimensions - each -X_i is already in span{X_i}. Therefore span dimension remains M.