How many vectors are perpendicular to \vec{a} =(-1, 1)? State the components of three such vectors.
For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.
a) \vec{a} = (-2, 1), \vec{b} = (1, 2)
For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.
b) \vec{a} = (2, 3,-1), \vec{b} = (4, 3, -17)
For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.
c) \vec{a} = (1, -2,5), \vec{b} = (3, -2, -2))
Give the components of a vector that is perpendicular to each of the following planes:
a) xy-plane
Give the components of a vector that is perpendicular to each of the following planes:
b) xz-plane
Give the components of a vector that is perpendicular to each of the following planes:
c) yz-plane
a) From the set of vectors \Big\{ (1, 2, 3), (-4, -5, -6), (4, 3, 10), (5, -3, \displaystyle{\frac{-5}{6}} \Big\}, select two pairs of vectors that are perpendicular to each other.
a) From the set of vectors \Big\{ (1, 2, 3), (-4, -5, -6), (4, 3, 10), (5, -3, \displaystyle{\frac{-5}{6}} \Big\}, select two pairs of vectors that are perpendicular to each other.
b) Are any oft these vectors collinear ? Explain.
a) Explain why it would not be possible to do this in \mathbb{R}^2 if we selected the two vectors \vec{a} = (1, -2) and \vec{b} = (1, 1).
b) Explain, in general, why it is not possible to do this if we select any two vectors in \mathbb{R}^2.
Determine the angle, to the nearest degree, between each of the following pairs of vectors:
a) \vec{a} = (5, 3) and \vec{b} = (-1, -2).
Determine the angle, to the nearest degree, between each of the following pairs of vectors:
b) \vec{a} = (-1, 4) and \vec{b} = (6, -2).
Determine the angle, to the nearest degree, between each of the following pairs of vectors:
c) \vec{a} = (2, 2, 1) and \vec{b} = (2, 1, -2).
Determine the angle, to the nearest degree, between each of the following pairs of vectors:
d) \vec{a} = (2, 3, -6) and \vec{b} = (-5, 0, 12).
Determine k, given two vectors and the angel between them.
a) \vec{a} = (-1, 2, -3) and \vec{b} = (-6k, -1, k), \theta = 90^o
Determine k, given two vectors and the angel between them.
b) \vec{a} = (1, 1) and \vec{b} = (0, k), \theta 45^o
In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.
a) Sketch the square.
In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.
b) Determine vector components for the two diagonals.
In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.
c) Verify that the angle between the diagonals is 90^o.
Determine the angle, to the nearest degree, between each pair of vectors.
\vec{a} = (1- \sqrt{2}, \sqrt{2} -1) and \vec{b} = (1 ,1)
Determine the angle, to the nearest degree, between each pair of vectors.
b) \vec{a} = (\sqrt{2} -1, \sqrt{2} +1, \sqrt{2}) and \vec{b} = (1 ,1, 1)
a) For the vectors \vec{a} = (2, p, 8) and \vec{b} = (q, 4, 12), determine values of p and q so that the actors are
i) collinear
ii) perpendicular
b) Are the values of p and q unique? Explain.
\triangle ABC has vertices at A(2, 5), B(4, 11), and C(-1, 6). Determine the angles in this triangle.
A rectangular box measuring 4 by 5 by 7 as when in the diagram at the left.
a) Determine the coordinates of each of the missing vertices.
A rectangular box measuring 4 by 5 by 7 as when in the diagram at the left.
b) Determine the angle, to the nearest degree, between \vec{AE} and \vec{BF}.
a) Given the vectors \vec{p} = (-1, 3, 0) and \vec{q} = (1, -5, 2), determine the components of a vector perpendicular to each of these vectors.
b) Given the vectors \vec{m} = (1, 3, -4) and \vec{n} = (-1 ,-2, 3), determine the components of a actor perpendicular to each of these vectors.
Find the value of p if the vectors \vec{r} = (p, p, 1) and \vec{s} = (p, -2, -3) are perpendicular to each other.
a) Determine the algebraic condition such that the vectors \vec{c} = (-3, p -1) and \vec{d} = (1, -4, q) are perpendicular to each other.
b) If q = -3, what is the corresponding value of p?
Given the actors \vec{r} = (1, 2, -1) and \vec{s} = (-2, -4, 2), determine the components of two veto perpendicular to each of these vectors. Explain you answer.
The vectors \vec{x} = (-4, p, -2) and \vec{y} = (-2, 3, 6) are such that \cos^{-1}(\displaystyle{\frac{4}{21}}) = \theta, where \theta is the angle between \vec{x} and \vec{y}. Determine the value(s) of p.
The diagonals of a parallelogram are determined by the vectors \vec{a} = 3, ,3 0) and \vec{b} = (-1, 1, -2).
a) Show that this parallelogram is a thumbs.
b) Determine vectors representing its sides and then determine the length of these sides.
The diagonals of a parallelogram are determined by the vectors \vec{a} = 3, ,3 0) and \vec{b} = (-1, 1, -2).
c) Determine the angles in this rhombus.
The rectangle ABCD has vertices at A(-1, 2, 3) , B(2, 6, -9), and D(3, q, 8).
a) Determine the coordinates of the vertex C.
b) Determine the angle between the two diagonals of this rectangle.
A cube measures 1 by 1 by 1. A line is drawn from one vertex to a diagonally opposite vertex through the centre of the cube. This is called a body diagonal for the cube. Determine the angles between the body diagonals of the cube.