Find λ for the points X(−1, 0), Y (1, λ) and Z(λ, 2), if angle YXZ = 90◦ (1 Viewer)

Haz_taz · Jan 8, 2021

Please help me with this question, i have tried to consider that the two gradients are XY and XZ, because when multiplied it should give -1, but λ cancels when doing so

username_2 · Jan 8, 2021

Haz_taz said:
Please help me with this question, i have tried to consider that the two gradients are XY and XZ, because when multiplied it should give -1, but λ cancels when doing so

This is best done using vector notation. Consider the vectors XY and XZ as given in the coordinates. Therefore:

XY = Y-X = (2,λ)
XZ = Z-X = (λ+1,2)

Hence, if we take the dot product of XY and XZ, we would get the following:
XY . XZ = (2 x (λ+1)) + (λ x 2) = 4λ + 2 = 0 (As a dot product of 0 indicates that the two vectors are perpendicular)

As a result, by rearranging the equation, we get that λ = - 1/2

Therefore, if angle YXZ is 90° then the value of λ is - 1/2.

TO verify using coordinate geo, let XY = λ/2 and XZ = 2/(λ+1)

Hence, by doing XY*XZ = λ/2 * 2/(λ+1) = λ/(λ+1) = -1

Therefore, λ = -λ - 1 => 2λ = -1 => therefore, λ = - 1/2

Correct me if I'm wrong.

Qeru · Jan 8, 2021

Haz_taz said:
Please help me with this question, i have tried to consider that the two gradients are XY and XZ, because when multiplied it should give -1, but λ cancels when doing so

Using gradients: For $YX\bot XZ$ we have $m_{YX}\times m_{ZX}=-1$ where m means the gradient. So $\frac{\lambda-0}{1-(-1)} \times \frac{2-0}{\lambda-(-1)}=-1$

$\frac{2\lambda}{2(\lambda+1)}=-1$

$\lambda=-(\lambda+1)$

$2\lambda=-1$

$\lambda=\frac{-1}{2}$

Oh just realised username already did that.

Haz_taz · Jan 8, 2021

Qeru said:
Using gradients: For $YX\bot XZ$ we have $m_{YX}\times m_{ZX}=-1$ where m means the gradient. So $\frac{\lambda-0}{1-(-1)} \times \frac{2-0}{\lambda-(-1)}=-1$

$\frac{2\lambda}{2(\lambda+1)}=-1$

$\lambda=-(\lambda+1)$

$2\lambda=-1$

$\lambda=\frac{-1}{2}$

Oh just realised username already did that.

Just understood it, THANKS

Haz_taz · Jan 8, 2021

username_2 said:
This is best done using vector notation. Consider the vectors XY and XZ as given in the coordinates. Therefore:

XY = Y-X = (2,λ)
XZ = Z-X = (λ+1,2)

Hence, if we take the dot product of XY and XZ, we would get the following:
XY . XZ = (2 x (λ+1)) + (λ x 2) = 4λ + 2 = 0 (As a dot product of 0 indicates that the two vectors are perpendicular)

As a result, by rearranging the equation, we get that λ = - 1/2

Therefore, if angle YXZ is 90° then the value of λ is - 1/2.

TO verify using coordinate geo, let XY = λ/2 and XZ = 2/(λ+1)

Hence, by doing XY*XZ = λ/2 * 2/(λ+1) = λ/(λ+1) = -1

Therefore, λ = -λ - 1 => 2λ = -1 => therefore, λ = - 1/2

Correct me if I'm wrong.

Ahhh, interesting approach, i understand now

Find λ for the points X(−1, 0), Y (1, λ) and Z(λ, 2), if angle YXZ = 90◦ (1 Viewer)

Haz_taz

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username_2

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Qeru

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Haz_taz

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Haz_taz

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