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Applications of Dot Product

Suppose the t...

Question

Suppose the three vectors $\to a, \to b, \to c$ on a plane satisfy the condition that $| \to a | = | \to b | = | \to c | = | \to a + \to b | = 1; \to c$ is perpendicular to $\to a$ and $\to b . \to c > 0,$ , then
(1)Find the angle formed by $2 \to a + \to b$ and $\to b$
(2)If the vector $\to c$ is expressed as a linear combination $λ \to a + μ \to b$ then find the ordered pair $(λ, μ)$

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Solution

$\begin{matrix} | \to a | = | \to b | = | \to c | = | \to a + \to b | = 1 \to a . \to c = 0 | \to a + \to b |^{2} = 1^{2} | \to a |^{2} + | \to b | + 2 | \to a | . | \to b | = 1 2 + 2 cos θ = 1 cos θ = \frac{1}{2} θ = 120^{\circ} N o w (2 \to a + \to b) . \to b = | 2 \to a + \to b | . | \to b | c o s α \sqrt{4 + 1 + 4 \times \frac{- 1}{2}} cos α = - 1 + 1 cos α = 0 s o, a n g l e, α = 90^{\circ} \to c = λ \to a + μ \to b \to c \to a = λ {\to a}^{2} + μ \to b \to a 0 = λ + μ \times (\frac{- 1}{2}) μ = 2 λ \to c = λ \to a + 2 λ \to b | \to c | = λ^{2} ({\to a}^{2} + 4 {\to b}^{2} + 4 (\frac{- 1}{2})) 1 = λ^{2} (1 + 4 - 2) λ = \frac{1}{\sqrt{3}} μ = \frac{2}{\sqrt{3}} (λ, μ) = (\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}) \end{matrix}$

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Similar questions

\to a, \to b, \to c

be three vectors such that

\to a \neq 0

\to a \times \to b = 2 \to a \times \to c, | \to a | = | \to c | = 1, | \to b | = 4

| \to b \times \to c | = \sqrt{15}

\to b - 2 \to c = λ \to a

λ

\to a, \to b, \to c

be three vectors such that

| \to a + \to b + \to c | = 1, \to c = λ (\to a \times \to b)

| \to a | = \frac{1}{\sqrt{2}}, | \to b | = \frac{1}{\sqrt{3}}, | \to c | = \frac{1}{\sqrt{6}}

, then the angle between

\to a

\to b

\to a

\to b

are perpendicular unit vectors and vector

\to c

\to c = \to a + \to b

(\to a \times \to b) . (\to b \times \to c) + (\to b \times \to c) . (\to c \times \to a) + (\to c \times \to a) . (\to a \times \to b)

\to a, \to b

\to c

\to a + \to b + \to c = \to 0

| \to a | = 3, | \to b = 5

| \to c | = 7

. Find the angle between

\to a

\to b .

\to a, \to b, \to c

are three non-zero vectors, no two of which are collinear and the vector

\to a + \to b

is collinear with

\to c, \to b + \to c

is collinear with

\to a,

\to a + \to b + \to c

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