Points and vectors

Dot Product of Vectors

Submitted by petr.beremlijski on Tue, 10/22/2024 - 13:41

2010015707

Level:

Given vectors

\vec{a} = (- 1; 2; 1)

\vec{b} = (0; - 1; 1)

, and

\vec{c} = (- 2; 0; 1)

, find the length of the vector

\vec{u} = \vec{a} - 2 \vec{b} + \vec{c}

| \vec{u} | = 5

| \vec{u} | = \sqrt{10}

| \vec{u} | = 3

| \vec{u} | = 1

2010015706

Level:

Given the vectors

\vec{a}

\vec{b}

, and

\vec{c}

, find

\vec{a} - 3 \vec{b} + \vec{c}

(10; 5)

(- 8; 5)

(- 8; - 1)

(7; 5)

2010015705

Level:

We are given points

A = [2; 1]

B = [7; 2]

, and

T = [4; 3]

, where point

T

is the centroid of triangle

A B C

. Find the coordinates of

C

, which is the vertex of

A B C

C = [3; 6]

C = [4; 8]

C = [3.5; 7]

C = [5; 6]

2010015704

Level:

Given the vectors

\vec{a}

\vec{b}

, and

\vec{c}

shown in the picture, express the vector

\vec{c}

as a linear combination of vectors

\vec{a}

and

\vec{b}

\vec{c} = - \vec{a} - 2 \vec{b}

\vec{c} = - \vec{a} + \frac{1}{2} \vec{b}

\vec{c} = - 2 \vec{a} - \vec{b}

\vec{c} = 2 \vec{a} + \frac{3}{2} \vec{b}

2010015703

Level:

The picture shows a rectangular cuboid

A B C D E F G H

. In the cuboid find the vector that is the sum of

\vec{A B} + \vec{A H} + \vec{E G} + \vec{F A} + \vec{H E}

\vec{A C}

\vec{F H}

\vec{A G}

\vec{B H}

2010015702

Level:

We are given the vectors

\vec{a} = (1; 2; - 3)

\vec{b} = (0; - 1; 2)

, and

\vec{c} = (- 1; 1; 0)

. Find the mixed product

(\vec{a} \times \vec{b}) \cdot \vec{c}

- 3

The mixed product is not defined.

(0; - 2; 0)

- 2

2010015701

Level:

S

is the midpoint of

A B

B = [3; - 5]

S = [0; - 7]

. Find the coordinates of

A

A = [- 3; - 9]

A = [1.5; - 6]

A = [3; - 12]

A = [- 3; - 2]

2010007410

Level:

Find the vector of length

\sqrt{2}

, which is perpendicular to the

y

-axis.

(\sqrt{2}; 0)

(2; 0)

(0; 2)

(0; \sqrt{2})

2010007409

Level:

Find the vector of length

\sqrt{3}

, which is perpendicular to the

x

-axis.

(0; \sqrt{3})

(3; 0)

(0; 3)

(\sqrt{3}; 0)

Dot Product of Vectors

2010015707

2010015706

2010015705

2010015704

2010015703

2010015702

2010015701

2010007410

2010007409

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